If you need to raise a specific number to a power, you can use. And now we will dwell on properties of degrees.

Exponential numbers they open up great possibilities, they allow us to transform multiplication into addition, and adding is much easier than multiplying.

For example, we need to multiply 16 by 64. The product of the multiplication of these two numbers is 1024. But 16 is 4x4, and 64 is 4x4x4. That is, 16 by 64 \u003d 4x4x4x4x4, which is also 1024.

The number 16 can also be represented as 2x2x2x2, and 64 as 2x2x2x2x2x2, and if we multiply, we again get 1024.

Now let's use the rule. 16 \u003d 4 2, or 2 4, 64 \u003d 4 3, or 2 6, at the same time 1024 \u003d 6 4 \u003d 4 5, or 2 10.

Therefore, our problem can be written differently: 4 2 x4 3 \u003d 4 5 or 2 4 x2 6 \u003d 2 10, and each time we get 1024.

We can solve a number of similar examples and see that multiplying numbers with powers reduces to addition of exponents, or exponential, of course, provided that the bases of the factors are equal.

Thus, without multiplying, we can immediately say that 2 4 x2 2 x2 14 \u003d 2 20.

This rule is also true when dividing numbers with powers, but in this case, e the exponent of the divisor is subtracted from the exponent of the dividend... Thus, 2 5: 2 3 \u003d 2 2, which in ordinary numbers is 32: 8 \u003d 4, that is, 2 2. Let's summarize:

a m х a n \u003d a m + n, a m: a n \u003d a m-n, where m and n are integers.

At first glance, it may seem what is multiplication and division of numbers with powers not very convenient, because first you need to represent the number in exponential form. It is not difficult to represent the numbers 8 and 16 in this form, that is, 2 3 and 2 4, but how to do this with the numbers 7 and 17? Or what to do when the number can be represented in exponential form, but the bases of the exponential expressions of numbers are very different. For example, 8 x 9 is 2 3 x 3 2, in which case we cannot sum the exponents. Neither 2 5 nor 3 5 is the answer, nor does the answer lie between these two numbers.

Then is it worth bothering with this method at all? Definitely worth it. It offers tremendous benefits, especially for complex and time-consuming calculations.

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Teaching aids and simulators in the Integral online store for grade 7
Manual for the textbook Yu.N. Makarycheva Manual for the textbook A.G. Mordkovich

The purpose of the lesson: learn how to perform actions with powers of number.

To begin with, let's remember the concept of "degree of number". An expression like $ \\ underbrace (a * a * \\ ldots * a) _ (n) $ can be represented as $ a ^ n $.

The converse is also true: $ a ^ n \u003d \\ underbrace (a * a * \\ ldots * a) _ (n) $.

This equality is called "notation of the degree as a product". It will help us determine how to multiply and divide degrees.
Remember:
a Is the base of the degree.
n - exponent.
If a n \u003d 1so the number and took once and accordingly: $ a ^ n \u003d 1 $.
If a n \u003d 0, then $ a ^ 0 \u003d 1 $.

Why this happens, we can figure out when we get acquainted with the rules of multiplication and division of powers.

Multiplication rules

a) If powers with the same base are multiplied.
To $ a ^ n * a ^ m $, write the degrees as a product: $ \\ underbrace (a * a * \\ ldots * a) _ (n) * \\ underbrace (a * a * \\ ldots * a) _ (m ) $.
The figure shows that the number and have taken n + m times, then $ a ^ n * a ^ m \u003d a ^ (n + m) $.

Example.
$2^3 * 2^2 = 2^5 = 32$.

It is convenient to use this property to simplify the work when raising a number to a large power.
Example.
$2^7= 2^3 * 2^4 = 8 * 16 = 128$.

b) If the degrees are multiplied with different bases, but the same exponent.
To $ a ^ n * b ^ n $, write the degrees as a product: $ \\ underbrace (a * a * \\ ldots * a) _ (n) * \\ underbrace (b * b * \\ ldots * b) _ (m ) $.
If we swap the multipliers and count the resulting pairs, we get: $ \\ underbrace ((a * b) * (a * b) * \\ ldots * (a * b)) _ (n) $.

So $ a ^ n * b ^ n \u003d (a * b) ^ n $.

Example.
$3^2 * 2^2 = (3 * 2)^2 = 6^2= 36$.

Division rules

a) The base of the degree is the same, the indicators are different.
Consider dividing an exponent with a larger exponent by dividing a exponent with a smaller exponent.

So it is necessary $ \\ frac (a ^ n) (a ^ m) $where n\u003e m.

Let's write the powers as a fraction:

$ \\ frac (\\ underbrace (a * a * \\ ldots * a) _ (n)) (\\ underbrace (a * a * \\ ldots * a) _ (m)) $.

For convenience, we will write the division as a simple fraction.

Now let's cancel the fraction.

It turns out: $ \\ underbrace (a * a * \\ ldots * a) _ (n-m) \u003d a ^ (n-m) $.
Hence, $ \\ frac (a ^ n) (a ^ m) \u003d a ^ (n-m) $.

This property will help explain the situation with raising a number to a zero power. Let us assume that n \u003d m, then $ a ^ 0 \u003d a ^ (n-n) \u003d \\ frac (a ^ n) (a ^ n) \u003d 1 $.

Examples.
$ \\ frac (3 ^ 3) (3 ^ 2) \u003d 3 ^ (3-2) \u003d 3 ^ 1 \u003d 3 $.

$ \\ frac (2 ^ 2) (2 ^ 2) \u003d 2 ^ (2-2) \u003d 2 ^ 0 \u003d 1 $.

b) The bases of the degree are different, the indicators are the same.
Let's say you need $ \\ frac (a ^ n) (b ^ n) $. Let's write the powers of numbers as a fraction:

$ \\ frac (\\ underbrace (a * a * \\ ldots * a) _ (n)) (\\ underbrace (b * b * \\ ldots * b) _ (n)) $.

For convenience, let's imagine.

Using the property of fractions, we split the large fraction into the product of small ones, we get.
$ \\ underbrace (\\ frac (a) (b) * \\ frac (a) (b) * \\ ldots * \\ frac (a) (b)) _ (n) $.
Accordingly: $ \\ frac (a ^ n) (b ^ n) \u003d (\\ frac (a) (b)) ^ n $.

Example.
$ \\ frac (4 ^ 3) (2 ^ 3) \u003d (\\ frac (4) (2)) ^ 3 \u003d 2 ^ 3 \u003d 8 $.

One of the main characteristics in algebra, and indeed in all mathematics, is the degree. Of course, in the 21st century, all calculations can be carried out on an online calculator, but it is better for the development of brains to learn how to do it yourself.

In this article, we will consider the most important questions regarding this definition. Namely, we will understand what it is in general and what are its main functions, what are the properties in mathematics.

Let's look at examples of what the calculation looks like, what are the basic formulas. Let's look at the main types of quantities and how they differ from other functions.

Let's understand how to solve various problems using this value. Let's show with examples how to raise to zero power, irrational, negative, etc.

Online Exponentiation Calculator

What is the degree of a number

What is meant by the expression "raise a number to a power"?

The power n of a is the product of multipliers of a value n times in a row.

Mathematically, it looks like this:

a n \u003d a * a * a *… a n.

For example:

• 2 3 \u003d 2 in the third step. \u003d 2 * 2 * 2 \u003d 8;
• 4 2 \u003d 4 in step. two \u003d 4 * 4 \u003d 16;
• 5 4 \u003d 5 in step. four \u003d 5 * 5 * 5 * 5 \u003d 625;
• 10 5 \u003d 10 in 5 steps. \u003d 10 * 10 * 10 * 10 * 10 \u003d 100000;
• 10 4 \u003d 10 in 4 steps. \u003d 10 * 10 * 10 * 10 \u003d 10000.

Below will be a table of squares and cubes from 1 to 10.

Grade table from 1 to 10

Below will be given the results of raising natural numbers to positive powers - "from 1 to 100".

Ch-lo	2nd article	3rd article
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	279
10	100	1000

Power properties

What is characteristic of such a mathematical function? Let's consider the basic properties.

Scientists have established the following signs characteristic of all degrees:

• a n * a m \u003d (a) (n + m);
• a n: a m \u003d (a) (n-m);
• (a b) m \u003d (a) (b * m).

Let's check with examples:

2 3 * 2 2 \u003d 8 * 4 \u003d 32.On the other hand 2 5 \u003d 2 * 2 * 2 * 2 * 2 \u003d 32.

Similarly: 2 3: 2 2 \u003d 8/4 \u003d 2. Otherwise 2 3 - 2 \u003d 2 1 \u003d 2.

(2 3) 2 \u003d 8 2 \u003d 64. And if it is different? 2 6 \u003d 2 * 2 * 2 * 2 * 2 * 2 \u003d 32 * 2 \u003d 64.

As you can see, the rules work.

But what about with addition and subtraction? It's simple. First, the exponentiation is performed, and only then the addition and subtraction.

Let's see some examples:

• 3 3 + 2 4 = 27 + 16 = 43;
• 5 2 - 3 2 \u003d 25 - 9 \u003d 16. Please note: the rule will not work if you subtract first: (5 - 3) 2 \u003d 2 2 \u003d 4.

But in this case, you must first calculate the addition, since there are actions in parentheses: (5 + 3) 3 \u003d 8 3 \u003d 512.

How to produce calculations in more complex cases? The procedure is the same:

• if there are brackets - you need to start with them;
• then exponentiation;
• then perform the actions of multiplication, division;
• after addition, subtraction.

There are specific properties that are not characteristic of all degrees:

1. The n-th root of the number a to the power of m will be written as: a m / n.
2. When raising a fraction to a power: both the numerator and its denominator are subject to this procedure.
3. When raising the product of different numbers to a power, the expression will correspond to the product of these numbers to a given power. That is: (a * b) n \u003d a n * b n.
4. When raising a number to a negative step., You need to divide 1 by a number in the same st-no, but with a "+" sign.
5. If the denominator of the fraction is in a negative power, then this expression will be equal to the product of the numerator and the denominator in the positive power.
6. Any number in degree 0 \u003d 1, and in step. 1 \u003d to yourself.

These rules are important in individual cases, we will consider them in more detail below.

Degree with negative exponent

What to do when the degree is minus, i.e. when the exponent is negative?

Based on properties 4 and 5 (see point above), turns out:

A (- n) \u003d 1 / A n, 5 (-2) \u003d 1/5 2 \u003d 1/25.

And vice versa:

1 / A (- n) \u003d A n, 1/2 (-3) \u003d 2 3 \u003d 8.

And if a fraction?

(A / B) (- n) \u003d (B / A) n, (3/5) (-2) \u003d (5/3) 2 \u003d 25/9.

Degree with a natural indicator

It is understood as a degree with indicators equal to whole numbers.

Things to remember:

A 0 \u003d 1, 1 0 \u003d 1; 2 0 \u003d 1; 3.15 0 \u003d 1; (-4) 0 \u003d 1 ... etc.

A 1 \u003d A, 1 1 \u003d 1; 2 1 \u003d 2; 3 1 \u003d 3 ... etc.

In addition, if (-a) 2 n +2, n \u003d 0, 1, 2 ... then the result will be with a "+" sign. If a negative number is raised to an odd power, then vice versa.

General properties, and all the specific features described above, are also characteristic of them.

Fractional degree

This view can be written by the scheme: A m / n. It reads as: n-th root of the number A to the m power.

You can do whatever you want with a fractional exponent: reduce it, decompose it into parts, raise it to a different degree, etc.

Irrational grade

Let α be an irrational number and A ˃ 0.

To understand the essence of a degree with such an indicator, consider different possible cases:

• A \u003d 1. The result will be equal to 1. Since there is an axiom - 1 in all degrees is equal to one;

А r 1 ˂ А α ˂ А r 2, r 1 ˂ r 2 - rational numbers;

• 0˂А˂1.

In this case, on the contrary: А r 2 ˂ А α ˂ А r 1 under the same conditions as in the second paragraph.

For example, the exponent is π. It is rational.

r 1 - in this case is equal to 3;

r 2 - will be equal to 4.

Then, for A \u003d 1, 1 π \u003d 1.

A \u003d 2, then 2 3 ˂ 2 π ˂ 2 4, 8 ˂ 2 π ˂ 16.

A \u003d 1/2, then (½) 4 ˂ (½) π ˂ (½) 3, 1/16 ˂ (½) π ˂ 1/8.

All the mathematical operations and specific properties described above are characteristic of these degrees.

Conclusion

To summarize - what are these values \u200b\u200bfor, what is the advantage of such functions? Of course, first of all, they simplify the life of mathematicians and programmers when solving examples, since they allow you to minimize calculations, shorten algorithms, organize data, and much more.

Where else can this knowledge be useful? In any working specialty: medicine, pharmacology, dentistry, construction, engineering, engineering, design, etc.

Expressions, expression conversion

Power expressions (expressions with powers) and their conversion

In this article, we will talk about converting power expressions. First, we will focus on transformations that are performed with expressions of any kind, including exponential expressions, such as expanding parentheses, casting similar terms. And then we will analyze the transformations inherent precisely in expressions with powers: working with the base and exponent, using the properties of degrees, etc.

Page navigation.

What are exponential expressions?

The term "exponential expressions" is practically not found in school textbooks of mathematics, but it appears quite often in collections of problems, especially those designed to prepare for the Unified State Exam and the OGE, for example,. After analyzing the tasks in which you need to perform any actions with exponential expressions, it becomes clear that expres- sions are understood as expressions containing powers in their records. Therefore, for yourself you can accept the following definition:

Definition.

Power expressions Are expressions containing degrees.

Let us give examples of power expressions... Moreover, we will represent them according to how the development of views on occurs from a degree with a natural indicator to a degree with a real indicator.

As you know, first there is an acquaintance with the power of a number with a natural exponent, at this stage the first simplest exponential expressions of the type 3 2, 7 5 +1, (2 + 1) 5, (−0,1) 4, 3 a 2 appear −a + a 2, x 3−1, (a 2) 3, etc.

A little later, the power of a number with an integer exponent is studied, which leads to the appearance of power expressions with negative integer powers, like the following: 3 −2, , a −2 + 2 b −3 + c 2.

In high school, they return to degrees again. There, a degree with a rational exponent is introduced, which entails the appearance of the corresponding power expressions: , etc. Finally, degrees with irrational indicators and expressions containing them are considered:,.

The matter is not limited to the listed power expressions: the variable penetrates further into the exponent, and, for example, such expressions 2 x 2 +1 or ... And after getting acquainted with, expressions with powers and logarithms begin to occur, for example, x 2 · lgx −5 · x lgx.

So, we figured out the question of what are exponential expressions. Next, we will learn how to transform them.

The main types of transformations of power expressions

With exponential expressions, you can perform any of the basic identical transformations of expressions. For example, you can expand parentheses, replace numeric expressions with their values, provide similar terms, etc. Naturally, in this case it is necessary to follow the accepted procedure for performing actions. Here are some examples.

Evaluate the value of the exponential expression 2 3 · (4 2 −12).

According to the order of performing the actions, we first perform the actions in brackets. There, firstly, we replace the degree 4 2 with its value 16 (see if necessary), and secondly, we calculate the difference 16−12 \u003d 4. We have 2 3 (4 2 −12) \u003d 2 3 (16−12) \u003d 2 3 4.

In the resulting expression, replace the power of 2 3 with its value 8, and then calculate the product 8 4 \u003d 32. This is the desired value.

So, 2 3 (4 2 - 12) \u003d 2 3 (16 - 12) \u003d 2 3 4 \u003d 8 4 \u003d 32.

2 3 (4 2 −12) \u003d 32.

Simplify Power Expressions 3 a 4 b −7 −1 + 2 a 4 b −7.

Obviously, this expression contains similar terms 3 · a 4 · b −7 and 2 · a 4 · b −7, and we can bring them:.

3 a 4 b −7 −1 + 2 a 4 b −7 \u003d 5 a 4 b −7 −1.

Imagine a product expression with powers.

To cope with the task, the representation of the number 9 as a power of 3 2 and the subsequent use of the formula for abbreviated multiplication is the difference of squares:

There are also a number of identical transformations inherent in power expressions. Then we will analyze them.

Working with the base and exponent

There are degrees, the base and / or exponent of which are not just numbers or variables, but some expressions. As an example, we present the entries (2 + 0.37) 5-3.7 and (a (a + 1) -a 2) 2 (x + 1).

When working with such expressions, you can replace both the expression based on the degree and the expression in the exponent with an identically equal expression on the ODZ of its variables. In other words, we can, according to the rules known to us, separately transform the base of the degree, and separately - the indicator. It is clear that as a result of this transformation, an expression is obtained that is identical to the original one.

Such transformations allow us to simplify expressions with powers or achieve other goals we need. For example, in the above exponential expression (2 + 0.3 · 7) 5-3.7, you can perform actions with the numbers in the base and exponent, which will allow you to go to the power 4.1 1.3. And after expanding the parentheses and reducing similar terms in the base of the degree (a (a + 1) −a 2) 2 (x + 1), we get a power expression of a simpler form a 2 (x + 1).

Using degree properties

One of the main tools for transforming expres- sions with powers is equalities reflecting. Let us recall the main ones. For any positive numbers a and b and arbitrary real numbers r and s, the following power properties are valid:

• a r a s \u003d a r + s;
• a r: a s \u003d a r − s;
• (a b) r \u003d a r b r;
• (a: b) r \u003d a r: b r;
• (a r) s \u003d a r s.

Note that for natural, integer, and also positive exponents, the restrictions on the numbers a and b may not be so strict. For example, for natural numbers m and n, the equality a m a n \u003d a m + n is true not only for positive a, but also for negative ones, and for a \u003d 0.

At school, the main attention when transforming power expressions is focused precisely on the ability to choose a suitable property and apply it correctly. In this case, the bases of degrees are usually positive, which allows using the properties of degrees without restrictions. The same applies to the transformation of expressions containing variables in the bases of degrees - the range of permissible values \u200b\u200bof variables is usually such that on it the bases take only positive values, which allows you to freely use the properties of degrees. In general, you need to constantly ask yourself whether it is possible in this case to apply any property of degrees, because inaccurate use of properties can lead to a narrowing of the ODV and other troubles. These points are discussed in detail and with examples in the article on conversion of expressions using degree properties. Here we restrict ourselves to a few simple examples.

Imagine the expression a 2.5 · (a 2) −3: a −5.5 as a power with base a.

First, we transform the second factor (a 2) −3 by the property of raising a power to a power: (a 2) −3 \u003d a 2 (−3) \u003d a −6... The original exponential expression will then take the form a 2.5 · a −6: a −5.5. Obviously, it remains to use the properties of multiplication and division of powers with the same base, we have
a 2.5 a −6: a −5.5 \u003d
a 2.5−6: a −5.5 \u003d a −3.5: a −5.5 \u003d
a −3.5 - (- 5.5) \u003d a 2.

a 2.5 (a 2) −3: a −5.5 \u003d a 2.

Power properties are used both from left to right and from right to left when transforming exponential expressions.

Find the value of the exponential expression.

Equality (a b) r \u003d a r b r, applied from right to left, allows you to go from the original expression to the product of the form and further. And when multiplying degrees with the same bases, the indicators add up: .

It was possible to perform the transformation of the original expression in another way:

.

Given the exponential expression a 1.5 −a 0.5 −6, enter the new variable t \u003d a 0.5.

The power of a 1.5 can be represented as a 0.5 · 3 and further, based on the property of the degree to the power (a r) s \u003d a r · s, applied from right to left, transform it to the form (a 0.5) 3. Thus, a 1.5 −a 0.5 −6 \u003d (a 0.5) 3 −a 0.5 −6... Now it is easy to introduce a new variable t \u003d a 0.5, we get t 3 −t − 6.

Converting fractions containing powers

Power expressions can contain fractions with powers or be such fractions. Any of the basic transformations of fractions that are inherent in fractions of any kind are fully applicable to such fractions. That is, fractions that contain powers can be canceled, reduced to a new denominator, worked separately with their numerator and separately with the denominator, etc. To illustrate the words spoken, consider solutions of several examples.

Simplify exponential expression .

This exponential expression is a fraction. Let's work with its numerator and denominator. In the numerator, we open the brackets and simplify the expression obtained after this, using the properties of the powers, and in the denominator we give similar terms:

And we also change the sign of the denominator by placing a minus in front of the fraction: .

.

The reduction of fractions containing powers to a new denominator is carried out similarly to the reduction of rational fractions to a new denominator. In this case, an additional factor is also found and the numerator and denominator of the fraction are multiplied by it. When performing this action, it is worth remembering that reduction to a new denominator can lead to a narrowing of the ODV. To prevent this from happening, it is necessary that the additional factor does not vanish for any values \u200b\u200bof the variables from the ODZ variables for the original expression.

Reduce fractions to a new denominator: a) to the denominator a, b) to the denominator.

a) In this case, it is quite easy to figure out which additional factor helps to achieve the desired result. This is a factor of a 0.3, since a 0.7 · a 0.3 \u003d a 0.7 + 0.3 \u003d a. Note that on the range of permissible values \u200b\u200bof the variable a (this is the set of all positive real numbers) the degree a 0.3 does not vanish, therefore, we have the right to multiply the numerator and denominator of the given fraction by this additional factor:

b) Looking more closely at the denominator, you can find that

and multiplying this expression by will give the sum of cubes and, that is,. And this is the new denominator to which we need to reduce the original fraction.

This is how we found an additional factor. On the range of admissible values \u200b\u200bof the variables x and y, the expression does not vanish, therefore, we can multiply the numerator and denominator of the fraction by it:

and) , b) .

The abbreviation of fractions containing powers is also nothing new: the numerator and denominator are represented as a number of factors, and the same factors of the numerator and denominator are canceled.

Reduce the fraction: a) , b).

a) First, the numerator and denominator can be reduced by the numbers 30 and 45, which is 15. Also, obviously, one can reduce by x 0.5 +1 and by ... Here's what we have:

b) In this case, the same factors in the numerator and denominator are not immediately visible. To get them, you will have to perform preliminary transformations. In this case, they consist in factoring the denominator into factors using the formula for the difference of squares:

and)

b) .

Reducing fractions to a new denominator and reducing fractions is mainly used to perform actions with fractions. Actions are performed according to known rules. When adding (subtracting) fractions, they are brought to a common denominator, after which the numerators are added (subtracted), and the denominator remains the same. The result is a fraction, the numerator of which is the product of the numerators, and the denominator is the product of the denominators. Division by a fraction is multiplication by the inverse of it.

Follow the steps .

First, we subtract the fractions in parentheses. To do this, we bring them to a common denominator, which is , after which we subtract the numerators:

Now we multiply the fractions:

Obviously, it is possible to cancel by a power of x 1/2, after which we have .

You can also simplify the exponential expression in the denominator by using the difference of squares formula: .

Simplify exponential expression .

Obviously, this fraction can be canceled by (x 2.7 +1) 2, this gives the fraction ... It is clear that something else needs to be done with the degrees of x. To do this, we transform the resulting fraction into a product. This allows us to use the property of dividing degrees with the same bases: ... And at the end of the process, we pass from the last product to a fraction.

.

And we also add that it is possible and in many cases desirable to transfer multipliers with negative exponents from the numerator to the denominator or from the denominator to the numerator, changing the sign of the exponent. Such transformations often simplify further actions. For example, an exponential expression can be replaced with.

Converting expressions with roots and powers

Often in expressions in which some transformations are required, along with powers with fractional exponents, there are also roots. To transform such an expression to the desired form, in most cases it is enough to go only to the roots or only to the powers. But since it is more convenient to work with degrees, they usually go from roots to degrees. However, it is advisable to carry out such a transition when the ODV of variables for the original expression allows you to replace the roots with powers without the need to refer to the module or split the ODV into several intervals (we discussed this in detail in the article the transition from roots to powers and back. a degree with an irrational indicator is introduced, which makes it possible to talk about a degree with an arbitrary real indicator.At this stage, the school begins to study exponential function, which is analytically set by the degree, at the base of which is the number, and in the indicator - the variable. So we are faced with exponential expressions containing numbers in the base of the degree, and in the exponent - expressions with variables, and naturally it becomes necessary to perform transformations of such expressions.

It should be said that the transformation of expressions of this type usually has to be performed when solving exponential equations and exponential inequalitiesand these conversions are pretty simple. In the overwhelming majority of cases, they are based on the properties of the degree and are mainly aimed at introducing a new variable in the future. We can demonstrate them by the equation 5 2 x + 1 −3 5 x 7 x −14 7 2 x − 1 \u003d 0.

First, the degrees, in which the sum of a variable (or expressions with variables) and a number is found, are replaced by products. This refers to the first and last terms of the expression on the left:
5 2 x 5 1 −3 5 x 7 x −14 7 2 x 7 −1 \u003d 0,
5 5 2 x −3 5 x 7 x −2 7 2 x \u003d 0.

Further, both sides of the equality are divided by the expression 7 2 x, which takes only positive values \u200b\u200bon the ODZ of the variable x for the original equation (this is a standard technique for solving equations of this kind, we are not talking about it now, so focus on the subsequent transformations of expressions with powers ):

Fractions with powers are now canceled, which gives .

Finally, the ratio of degrees with the same exponents is replaced by the degrees of relations, which leads to the equation which is equivalent ... The performed transformations allow us to introduce a new variable, which reduces the solution of the original exponential equation to solving the quadratic equation

I. V. Boykov, L. D. Romanova Collection of tasks for preparing for the exam. Part 1. Penza 2003.

Sections: Maths

Lesson type: lesson in generalization and systematization of knowledge

Objectives:

teaching - repeat the definition of the degree, the rules for multiplying and dividing degrees, raising a degree to a power, consolidate the skills of solving examples containing degrees,

developing - development of logical thinking of students, interest in the studied material,

educating - fostering a responsible attitude towards learning, a culture of communication, a sense of collectivism.

Equipment: computer, multimedia projector, interactive whiteboard, presentation “Degrees” for oral counting, cards with assignments, handouts.

Lesson plan:

Organizing time.

Repeating rules

Verbal counting.

Historical reference.

Work at the blackboard.

Physical education.

Work on an interactive whiteboard.

Independent work.

Homework.

Summing up the lesson.

During the classes

I. Organizational moment

Communication of the topic and objectives of the lesson.

In the previous lessons, you discovered the wonderful world of degrees, learned how to multiply and divide degrees, raise them to a degree. Today we must consolidate the knowledge gained by solving examples.

II. Repeating rules (orally)

1. Give a definition of a degree with a natural indicator? (By the power of the number and with a natural exponent greater than 1 is called the product n factors, each of which is equal to and.)
2. How to multiply two degrees? (To multiply powers with the same bases, you must leave the base the same and add the exponents.)
3. How do you divide a degree by a degree? (To divide degrees with the same bases, you must leave the base the same, and subtract the indicators.)
4. How to raise a work to a power? (To raise a product to a power, you need to raise each factor to this power)
5. How to raise a degree to a degree? (To raise a power to a power, you need to leave the base the same, and multiply the indicators)

III. Verbal counting (by multimedia)

IV. Historical reference

All tasks are from the Ahmes papyrus, which was written about 1650 BC. e. related to the practice of construction, delimitation of land plots, etc. The tasks are grouped by topic. For the most part, these are tasks for finding the areas of a triangle, quadrangles and a circle, various actions with integers and fractions, proportional division, finding relations, there is also raising to different powers, solving equations of the first and second degrees with one unknown.

There is no explanation or proof whatsoever. The desired result is either given directly, or a short algorithm for its calculation is given. This way of presentation, typical for the science of the countries of the ancient East, suggests that mathematics there developed by means of generalizations and guesses that did not form any general theory. However, the papyrus contains a number of evidence that Egyptian mathematicians knew how to root and exponentiate, solve equations, and even owned the rudiments of algebra.

V. Working at the blackboard

Find the meaning of the expression in a rational way:

Calculate the value of the expression:



Vi. Physical education

6. for eyes
7. for neck
8. for hands
9. for the torso
10. for legs

Vii. Solving problems (shown on an interactive whiteboard)

Is the root of the equation a positive number?

xn - i1abbnckbmcl9fb.xn - p1ai

Formulas for degrees and roots.

Power formulas are used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.

Number c is an npower of the number a when:



Operations with degrees.

1. Multiplying degrees with the same base, their indicators add up:

2. In the division of degrees with the same base, their indicators are subtracted:



3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:

(abc ...) n \u003d a n b n c n ...

4. The power of the fraction is equal to the ratio of the powers of the dividend and the divisor:

5. Raising a degree to a degree, the exponents are multiplied:

Each of the above formula is true in the left-to-right directions and vice versa.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of the relationship is equal to the ratio of the dividend and the divisor of the roots:



3. When raising a root to a power, it is enough to raise the root number to this power:



4. If you increase the degree of the root in n once and at the same time to build n-th power of the root number, the root value will not change:



5. If you reduce the degree of the root in n extract the root once and at the same time n-th power of the root number, then the root value will not change:



The power of a number with a non-positive (whole) exponent is defined as a unit divided by the power of the same number with an exponent equal to the absolute value of the non-positive exponent:



Formula a m : a n \u003d a m - n can be used not only for m > n , but also at m 4: a 7 \u003d a 4 - 7 \u003d a -3.

So that the formula a m : a n \u003d a m - n became fair when m \u003d n, you need the presence of zero degree.

The power of any nonzero number with zero exponent equals one.

To erect a real number and to the extent m / n, you need to extract the root n-Th degree of m-th power of this number and:



Degree formulas.

6. a - n = - division of degrees;

7. - division of degrees;

8.a 1 / n \u003d ;

Degrees rule of action with powers

1. The degree of the product of two or more factors is equal to the product of the degrees of these factors (with the same exponent):

(abc ...) n \u003d a n b n c n ...

Example 1. (7 2 10) 2 \u003d 7 2 2 2 10 2 \u003d 49 4 100 \u003d 19600. Example 2. (x 2 –a 2) 3 \u003d [(x + a) (x - a)] 3 \u003d ( x + a) 3 (x - a) 3

In practical terms, the reverse is more important:

a n b n c n… \u003d (abc…) n

those. the product of the same powers of several quantities is equal to the same power of the product of these quantities.

Example 3. Example 4. (a + b) 2 (a 2 - ab + b 2) 2 \u003d [(a + b) (a 2 - ab + b 2)] 2 \u003d (a 3 + b 3) 2

2. The power of the quotient (fraction) is equal to the quotient of dividing the same power of the divisor by the same power of the divisor:

Example 5. Example 6.

Reverse conversion: Example 7. ... Example 8. .

3. When multiplying degrees with the same bases, the exponents are added:

Example 9.2 2 2 5 \u003d 2 2 + 5 \u003d 2 7 \u003d 128. Example 10 (a - 4c + x) 2 (a - 4c + x) 3 \u003d (a - 4c + x) 5.

4. When dividing degrees with the same bases, the exponent of the divisor is subtracted from the exponent of the dividend

Example 11.12 5:12 3 \u003d 12 5-3 \u003d 12 2 \u003d 144. Example 12 (x-y) 3: (x-y) 2 \u003d x-y.

5. When raising a degree to a power, the exponents are multiplied:

Example 13. (2 3) 2 \u003d 2 6 \u003d 64. Example 14.

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Degrees and roots

Operations with powers and roots. Degree with negative ,

zero and fractional indicator. About expressions that don't make sense.

Operations with degrees.

1. When multiplying degrees with the same base, their indicators are added:

a m · a n \u003d a m + n.

2. When dividing degrees with the same base, their indicators deducted .



3. The degree of the product of two or more factors is equal to the product of the degrees of these factors.

4. The degree of the ratio (fraction) is equal to the ratio of the degrees of the dividend (numerator) and divisor (denominator):

(a / b) n \u003d a n / b n.

5. When raising a degree to a degree, their indicators are multiplied:

All of the above formulas are read and executed in both directions from left to right and vice versa.

PRI me r. (2 · 3 · 5/15) ² = 2 ² 3 ² 5 ² / 15 ² \u003d 900/225 \u003d 4 .

Operations with roots. In all the formulas below, the symbol means arithmetic root (the radical expression is positive).

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of the ratio is equal to the ratio of the roots of the dividend and the divisor:



3. When raising a root to a power, it is enough to raise to this power root number:



4. If we increase the degree of the root by m times and at the same time raise the radical number to the m-th power, then the value of the root will not change:



5. If we reduce the degree of the root by m times and at the same time extract the mth root from the radical number, then the value of the root will not change:





Expansion of the concept of degree. Until now, we have considered degrees only with a natural exponent; but actions with powers and roots can also lead to negative, zero and fractional indicators. All these degree indicators require additional definition.

Degree with negative exponent. The power of a number with a negative (integer) exponent is defined as a unit divided by the power of the same number with an exponent equal to the absolute value of a negative exponent:



Now the formula a m : a n = a m - n can be used not only for m greater than n , but also at m less than n .

PRI me r. a 4: a 7 \u003d a 4 - 7 \u003d a - 3 .

If we want the formula a m : a n = a m - n was fair when m \u003d n , we need a definition of the zero degree.

Zero degree. The power of any nonzero number with exponent zero is 1.

EXAMPLES 2 0 \u003d 1, ( – 5) 0 = 1, (– 3 / 5) 0 = 1.

Fractional exponent. In order to raise a real number a to the power of m / n, you need to extract the nth root of the mth power of this number a:



About expressions that don't make sense. There are several such expressions.

where a ≠ 0 , does not exist.

Indeed, assuming that x - some number, then in accordance with the definition of the division operation we have: a = 0· x, i.e. a \u003d 0, which contradicts the condition: a ≠ 0

- any number.

Indeed, if we assume that this expression is equal to some number x, then according to the definition of the division operation we have: 0 \u003d 0 x ... But this equality holds for any number x, as required to prove.

0 0 - any number.



Solution. Consider three main cases:

1) x = 0 – this value does not satisfy this equation

2) at x \u003e 0 we get: x / x \u003d 1, i.e. 1 \u003d 1, whence it follows

what x - any number; but taking into account that in

our case x \u003e 0, the answer is x > 0 ;

Degree properties

We remind you that this lesson understands power properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in the lessons for grade 8.

A natural exponent has several important properties that make it easier to calculate in exponent examples.

Property number 1
Product of degrees

When multiplying degrees with the same bases, the base remains unchanged, and the exponents are added.

a m · a n \u003d a m + n, where "a" is any number, and "m", "n" are any natural numbers.

This property of degrees also affects the product of three or more degrees.

• Simplify the expression.
  b b 2 b 3 b 4 b 5 \u003d b 1 + 2 + 3 + 4 + 5 \u003d b 15
• Present as a degree.
  6 15 36 \u003d 6 15 6 2 \u003d 6 15 6 2 \u003d 6 17
• Present as a degree.
  (0.8) 3 (0.8) 12 \u003d (0.8) 3 + 12 \u003d (0.8) 15

Please note that in the specified property it was only about the multiplication of powers with the same bases ... It does not apply to their addition.

You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if
count (3 3 + 3 2) \u003d (27 + 9) \u003d 36, and 3 5 \u003d 243

Property number 2
Private degrees

When dividing degrees with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

• Write the quotient as a degree
  (2b) 5: (2b) 3 \u003d (2b) 5 - 3 \u003d (2b) 2
• Calculate.

11 3 - 2 4 2 - 1 \u003d 11 4 \u003d 44
Example. Solve the equation. We use the property of private degrees.
3 8: t \u003d 3 4

Answer: t \u003d 3 4 \u003d 81

Using properties # 1 and # 2, you can easily simplify expressions and perform calculations.

Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 \u003d 4 5m + 6 + m + 2: 4 4m + 3 \u003d 4 6m + 8 - 4m - 3 \u003d 4 2m + 5

Example. Find the value of an expression using the properties of the degree.

2 11 − 5 = 2 6 = 64

Note that property 2 was only about dividing degrees with the same bases.

The difference (4 3 −4 2) cannot be replaced with 4 1. This is understandable if we calculate (4 3 −4 2) \u003d (64 - 16) \u003d 48, and 4 1 \u003d 4

Property number 3
Exponentiation

When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.

(a n) m \u003d a n · m, where "a" is any number, and "m", "n" are any natural numbers.

Example.
(a 4) 6 \u003d a 4 6 \u003d a 24

Example. Represent 3 20 as a power with base 3 2.

By the property of raising a power to a power it is known that when raised to a power, the indicators are multiplied, which means:

Properties 4
Work degree

When raising a power to a power of a product, each factor is raised to this power and the results are multiplied.

(a · b) n \u003d a n · b n, where "a", "b" are any rational numbers; "N" is any natural number.

• Example 1.
  (6 a 2 b 3 s) 2 \u003d 6 2 a 2 2 b 3 2 s 1 2 \u003d 36 a 4 b 6 s 2
• Example 2.
  (−x 2 y) 6 \u003d ((−1) 6 x 2 6 y 1 6) \u003d x 12 y 6

Note that property # 4, like other degree properties, is applied in reverse order.

(a n b n) \u003d (a b) n

That is, in order to multiply degrees with the same indicators, you can multiply the bases, and the exponent can be left unchanged.

• Example. Calculate.
  2 4 5 4 \u003d (2 5) 4 \u003d 10 4 \u003d 10,000
• Example. Calculate.
  0.5 16 2 16 \u003d (0.5 2) 16 \u003d 1

In more complex examples, there may be cases when multiplication and division must be performed over degrees with different bases and different exponents. In this case, we advise you to proceed as follows.

For example, 4 5 3 2 \u003d 4 3 4 2 3 2 \u003d 4 3 (4 3) 2 \u003d 64 12 2 \u003d 64 144 \u003d 9216

An example of raising to a decimal power.

4 21 (−0.25) 20 \u003d 4 4 20 (−0.25) 20 \u003d 4 (4 (−0.25)) 20 \u003d 4 (−1) 20 \u003d 4 1 \u003d 4

Properties 5
Degree of quotient (fraction)

To raise a quotient to a power, you can raise a separate dividend and a divisor to this power, and divide the first result by the second.

(a: b) n \u003d a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n is any natural number.

• Example. Present the expression in the form of private degrees.
  (5: 3) 12 = 5 12: 3 12

We remind you that the quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.

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To begin with, let's recall the basic formulas of degrees and their properties.

Product of number a happens to itself n times, we can write this expression as a a ... a \u003d a n

1.a 0 \u003d 1 (a ≠ 0)

3.a n a m \u003d a n + m

4. (a n) m \u003d a nm

5.a n b n \u003d (ab) n

7.a n / a m \u003d a n - m

Power or exponential equations - these are equations in which the variables are in powers (or exponents), and the base is a number.

Examples of exponential equations:

In this example, the number 6 is the base, it always stands at the bottom, and the variable x degree or indicator.

Here are some more examples of exponential equations.
2 x * 5 \u003d 10
16 x - 4 x - 6 \u003d 0

Now let's see how the exponential equations are solved?

Let's take a simple equation:

2 x \u003d 2 3

Such an example can be solved even in the mind. It is seen that x \u003d 3. After all, in order for the left and right sides to be equal, you need to put the number 3 instead of x.
Now let's see how this solution needs to be formalized:

2 x \u003d 2 3
x \u003d 3

In order to solve such an equation, we removed identical grounds (that is, two's) and wrote down what was left, these are degrees. We got the desired answer.

Now let's summarize our decision.

Algorithm for solving the exponential equation:
1. Need to check the same whether the equation has bases on the right and left. If the grounds are not the same, we are looking for options to solve this example.
2. After the bases are the same, equate degree and solve the resulting new equation.

Now let's solve a few examples:

Let's start simple.

The bases on the left and right sides are equal to the number 2, so we can discard the base and equate their degrees.

x + 2 \u003d 4 This is the simplest equation.
x \u003d 4 - 2
x \u003d 2
Answer: x \u003d 2

In the following example, you can see that the bases are different - 3 and 9.

3 3x - 9x + 8 \u003d 0

First, we move the nine to the right side, we get:

Now you need to make the same bases. We know that 9 \u003d 3 2. Let's use the formula of degrees (a n) m \u003d a nm.

3 3x \u003d (3 2) x + 8

We get 9 x + 8 \u003d (3 2) x + 8 \u003d 3 2x + 16

3 3x \u003d 3 2x + 16 now you can see that the bases on the left and right sides are the same and equal to three, so we can discard them and equate the degrees.

3x \u003d 2x + 16 got the simplest equation
3x - 2x \u003d 16
x \u003d 16
Answer: x \u003d 16.

See the following example:

2 2x + 4 - 10 4 x \u003d 2 4

First of all, we look at the bases, the bases are different two and four. And we need to be - the same. We transform the four by the formula (a n) m \u003d a nm.

4 x \u003d (2 2) x \u003d 2 2x

And we also use one formula a n a m \u003d a n + m:

2 2x + 4 \u003d 2 2x 2 4

Add to the equation:

2 2x 2 4 - 10 2 2x \u003d 24

We have led the example to the same grounds. But we are hindered by other numbers 10 and 24. What to do with them? If you look closely, you can see that 2 2x is repeated on the left side, here is the answer - 2 2x we can take out of the brackets:

2 2x (2 4 - 10) \u003d 24

Let's calculate the expression in brackets:

2 4 - 10 = 16 - 10 = 6

Divide the whole equation by 6:

Let's imagine 4 \u003d 2 2:

2 2x \u003d 2 2 bases are the same, discard them and equate the powers.
2x \u003d 2 it turns out the simplest equation. We divide it by 2 we get
x \u003d 1
Answer: x \u003d 1.

Let's solve the equation:

9 x - 12 * 3 x + 27 \u003d 0

Let's transform:
9 x \u003d (3 2) x \u003d 3 2x

We get the equation:
3 2x - 12 3x +27 \u003d 0

Our bases are the same equal to 3. In this example, you can see that the first three has a degree twice (2x) than the second (just x). In this case, you can solve replacement method... We replace the number with the smallest degree:

Then 3 2x \u003d (3x) 2 \u003d t 2

Replace all powers with x in the equation with t:

t 2 - 12t + 27 \u003d 0
We get a quadratic equation. We solve through the discriminant, we get:
D \u003d 144-108 \u003d 36
t 1 \u003d 9
t 2 \u003d 3

Back to the variable x.

We take t 1:
t 1 \u003d 9 \u003d 3 x

That is,

3 x \u003d 9
3 x \u003d 3 2
x 1 \u003d 2

One root was found. We are looking for the second, from t 2:
t 2 \u003d 3 \u003d 3 x
3 x \u003d 3 1
x 2 \u003d 1
Answer: x 1 \u003d 2; x 2 \u003d 1.

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The concept of a degree in mathematics is introduced in the 7th grade at the algebra lesson. And in the future, throughout the course of studying mathematics, this concept is actively used in its various forms. Degrees are a rather difficult topic that requires memorizing the meanings and the ability to correctly and quickly count. For faster and better work with degrees, mathematicians invented the properties of the degree. They help to cut down on big computations, to convert a huge example to one number to some extent. There are not so many properties, and all of them are easy to remember and apply in practice. Therefore, the article discusses the main properties of the degree, as well as where they are applied.

Degree properties

We will consider 12 properties of a degree, including properties of degrees with the same bases, and give an example for each property. Each of these properties will help you solve problems with degrees faster, as well as save you from numerous computational errors.

1st property.

Many people very often forget about this property, make mistakes, representing a number in the zero degree as zero.

2nd property.

3rd property.

It must be remembered that this property can only be applied when multiplying numbers, it does not work with a sum! And we must not forget that this, and the next, properties apply only to degrees with the same bases.

4th property.

If the number in the denominator is raised to a negative power, then during subtraction, the power of the denominator is taken in parentheses for the correct sign change in further calculations.

The property works only for division, it does not apply for subtraction!

5th property.

6th property.

This property can be applied in the opposite direction. The unit divided by the number is to some extent this number in the minus power.

7th property.

This property cannot be applied to sum and difference! When raising a sum or difference to a power, abbreviated multiplication formulas are used, not power properties.

8th property.

9th property.

This property works for any fractional power with a numerator equal to one, the formula will be the same, only the power of the root will change depending on the denominator of the power.

Also, this property is often used in reverse order. The root of any power of a number can be represented as the number to the power of one divided by the power of the root. This property is very useful in cases where the root of a number is not extracted.

10th property.

This property works for more than just square root and second degree. If the degree of the root and the degree to which this root is raised coincide, then the answer will be a radical expression.

11th property.

You need to be able to see this property in time when making a decision in order to save yourself from huge calculations.

12th property.

Each of these properties will come across you more than once in assignments, it can be given in its pure form, or it may require some transformations and the use of other formulas. Therefore, for the correct solution, it is not enough to know only the properties, you need to practice and connect the rest of the mathematical knowledge.

Applying degrees and their properties

They are actively used in algebra and geometry. Degrees in mathematics have a separate, important place. With their help, exponential equations and inequalities are solved, as well as degrees of often complicate equations and examples related to other branches of mathematics. Degrees help to avoid large and time-consuming calculations, degrees are easier to abbreviate and calculate. But to work with large degrees, or with powers of large numbers, you need to know not only the properties of the degree, but also to work competently with the bases, to be able to decompose them in order to facilitate your task. For convenience, you should also know the meaning of the numbers raised to a power. This will shorten your decision time, eliminating the need for long calculations.

The concept of degree plays a special role in logarithms. Since the logarithm, in essence, is the power of a number.

Abbreviated multiplication formulas are another example of the use of powers. The properties of degrees cannot be applied in them, they are decomposed according to special rules, but degrees are invariably present in each formula for abbreviated multiplication.

Degrees are also actively used in physics and computer science. All translations into the SI system are made using degrees, and later, when solving problems, the properties of the degree are applied. In computer science, powers of two are actively used, for the convenience of counting and simplifying the perception of numbers. Further calculations for conversions of units of measurement or calculations of problems, as in physics, are performed using the properties of the degree.

Degrees are also very useful in astronomy, where you rarely find the use of the properties of the degree, but the degrees themselves are actively used to shorten the recording of various quantities and distances.

Degrees are also used in everyday life, when calculating areas, volumes, distances.

With the help of degrees, very large and very small values \u200b\u200bare recorded in all areas of science.

Exponential equations and inequalities

It is in exponential equations and inequalities that the properties of degree occupy a special place. These tasks are very common, both in the school course and in exams. All of them are solved by applying the properties of the degree. The unknown is always in the degree itself, so knowing all the properties, it will not be difficult to solve such an equation or inequality.

In the previous article, we described what monomials are. In this article, we will analyze how to solve examples and problems in which they are applied. Here we will consider such actions as subtraction, addition, multiplication, division of monomials and raising them to a power with a natural exponent. We will show how such operations are defined, outline the basic rules for their implementation and what should be the result. All theoretical positions, as usual, will be illustrated with examples of problems with descriptions of solutions.

It is most convenient to work with the standard notation of monomials, therefore, all expressions that will be used in the article are presented in a standard form. If initially they are set differently, it is recommended to first bring them to the generally accepted form.

Addition and Subtraction Rules for Monomials

The simplest operations that can be performed with monomials are subtraction and addition. In the general case, the result of these actions will be a polynomial (a monomial is possible in some special cases).

When we add or subtract monomials, we first write down the corresponding sum and difference in conventional form, and then simplify the resulting expression. If there are such terms, they need to be given, brackets - to open. Let us explain with an example.

Example 1

Condition: perform the addition of monomials - 3 x and 2, 72 x 3 y 5 z.

Decision

Let's write down the sum of the original expressions. Let's add parentheses and put a plus between them. We get the following:

(- 3 x) + (2, 72 x 3 y 5 z)

When we expand the parentheses, we get - 3 x + 2.72 x 3 y 5 z. This is a polynomial written in standard form, which will be the result of the addition of these monomials.

Answer: (- 3 x) + (2, 72 x 3 y 5 z) \u003d - 3 x + 2, 72 x 3 y 5 z.

If we have three, four or more terms, we carry out this action in the same way.

Example 2

Condition: perform the indicated actions with the polynomials in the correct order

3 a 2 - (- 4 a c) + a 2 - 7 a 2 + 4 9 - 2 2 3 a c

Decision

Let's start by expanding the brackets.

3 a 2 + 4 a c + a 2 - 7 a 2 + 4 9 - 2 2 3 a c

We see that the resulting expression can be simplified by bringing similar terms:

3 a 2 + 4 a c + a 2 - 7 a 2 + 4 9 - 2 2 3 a c \u003d \u003d (3 a 2 + a 2 - 7 a 2) + 4 a c - 2 2 3 a c + 4 9 \u003d \u003d - 3 a 2 + 1 1 3 a c + 4 9

We have got a polynomial, which will be the result of this action.

Answer: 3 a 2 - (- 4 a c) + a 2 - 7 a 2 + 4 9 - 2 2 3 a c \u003d - 3 a 2 + 1 1 3 a c + 4 9

In principle, we can add and subtract two monomials with some restrictions so that we end up with a monomial. To do this, you need to comply with some conditions concerning the terms and subtracted monomials. We will describe how this is done in a separate article.

Multiplication rules for monomials

The multiplication action does not impose any restrictions on the multipliers. The monomials to be multiplied do not have to meet any additional conditions for the result to be a monomial.

To perform the multiplication of monomials, you need to follow these steps:

1. Record the piece correctly.
2. Expand parentheses in the resulting expression.
3. Group, if possible, factors with the same variables and numerical factors separately.
4. Perform the necessary actions with the numbers and apply the property of multiplying powers with the same bases to the remaining factors.

Let's see how this is done in practice.

Example 3

Condition: multiply the monomials 2 x 4 y z and - 7 16 t 2 x 2 z 11.

Decision

Let's start by compiling a work.

We open the brackets in it and get the following:

2 x 4 y z - 7 16 t 2 x 2 z 11

2 - 7 16 t 2 x 4 x 2 y z 3 z 11

All we have left to do is multiply the numbers in the first brackets and apply the power property for the second. As a result, we get the following:

2 - 7 16 t 2 x 4 x 2 y z 3 z 11 \u003d - 7 8 t 2 x 4 + 2 y z 3 + 11 \u003d \u003d - 7 8 t 2 x 6 y z 14

Answer: 2 x 4 y z - 7 16 t 2 x 2 z 11 \u003d - 7 8 t 2 x 6 y z 14.

If we have three or more polynomials in our condition, we multiply them according to exactly the same algorithm. We will consider in more detail the question of multiplication of monomials in a separate material.

Rules for raising a monomial to a power

We know that a degree with a natural exponent is the product of a certain number of identical factors. Their number is indicated by the number in the indicator. According to this definition, raising a monomial to a power is equivalent to multiplying the specified number of identical monomials. Let's see how this is done.

Example 4

Condition: raise the monomial - 2 a b 4 to the power of 3.

Decision

We can replace exponentiation with multiplication of 3 monomials - 2 · a · b 4. Let's write down and get the desired answer:

(- 2 a b 4) 3 \u003d (- 2 a b 4) (- 2 a b 4) (- 2 a b 4) \u003d \u003d ((- 2) (- 2) (- 2)) (a a a) (b 4 b 4 b 4) \u003d - 8 a 3 b 12

Answer: (- 2 a b 4) 3 \u003d - 8 a 3 b 12.

But what if the degree has a large indicator? Writing a large number of factors is inconvenient. Then, to solve such a problem, we need to apply the properties of the degree, namely the property of the degree of the product and the property of the degree in degree.

Let's solve the problem that we gave above in the indicated way.

Example 5

Condition: perform the erection - 2 · a · b 4 to the third power.

Decision

Knowing the property of the degree to the degree, we can proceed to an expression of the following form:

(- 2 a b 4) 3 \u003d (- 2) 3 a 3 (b 4) 3.

After that we raise to the power - 2 and apply the power property to the power:

(- 2) 3 (a) 3 (b 4) 3 \u003d - 8 a 3 b 4 3 \u003d - 8 a 3 b 12.

Answer: - 2 a b 4 \u003d - 8 a 3 b 12.

We also devoted a separate article to raising a monomial to a power.

Division rules for monomials

The last action with monomials, which we will analyze in this material, is the division of a monomial by a monomial. As a result, we should get a rational (algebraic) fraction (in some cases it is possible to obtain a monomial). Let us clarify right away that division by a zero monomial is not defined, since division by 0 is not defined.

To perform division, we need to write down the indicated monomials in the form of a fraction and reduce it, if possible.

Example 6

Condition: perform the division of the monomial - 9 x 4 y 3 z 7 by - 6 p 3 t 5 x 2 y 2.

Decision

Let's start by writing monomials in fractional form.

9 x 4 y 3 z 7 - 6 p 3 t 5 x 2 y 2

This fraction can be reduced. After performing this action, we get:

3 x 2 y z 7 2 p 3 t 5

Answer: - 9 x 4 y 3 z 7 - 6 p 3 t 5 x 2 y 2 \u003d 3 x 2 y z 7 2 p 3 t 5.

The conditions under which, as a result of division of monomials, we obtain a monomial are given in a separate article.

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How to multiply degrees? Which degrees can be multiplied and which cannot? How to multiply the number by the degree?

In algebra, the product of degrees can be found in two cases:

1) if the degrees have the same bases;

2) if the degrees have the same indicators.

When multiplying degrees with the same bases, the base must be left the same, and the indicators must be added:

When multiplying degrees with the same indicators, the total indicator can be taken out of the brackets:

Let's look at how to multiply degrees using specific examples.

The unit in the exponent is not written, but when the degrees are multiplied, they take into account:

When multiplying, the number of degrees can be any. It should be remembered that you don't have to write the multiplication sign before the letter:

In expressions, exponentiation is performed first.

If you need to multiply a number by a power, you must first perform the exponentiation, and only then the multiplication:

www.algebraclass.ru

Addition, subtraction, multiplication, and division of powers

Add and subtract powers

Obviously, numbers with powers can be added, like other quantities , by adding them one by one with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.

Odds equal degrees of the same variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is 5a 2.

It is also obvious that if you take two squares a, or three squares a, or five squares a.

But degrees different variables and varying degrees identical variables, must be added by their addition with their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3.

It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.

Subtraction degrees is carried out in the same way as addition, except that the signs of the subtracted must be changed accordingly.

Or:
2a 4 - (-6a 4) \u003d 8a 4
3h 2 b 6 - 4h 2 b 6 \u003d -h 2 b 6
5 (a - h) 6 - 2 (a - h) 6 \u003d 3 (a - h) 6

Multiplication of degrees

Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without the multiplication sign between them.

So, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m \u003d a m x -3
3a 6 y 2 ⋅ (-2x) \u003d -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y \u003d a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding the same variables.
The expression will take the form: a 5 b 5 y 3.

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to sum degrees of terms.

So, a 2 .a 3 \u003d aa.aaa \u003d aaaaa \u003d a 5.

Here 5 is the power of the result of multiplication, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m \u003d a m + n.

For a n, a is taken as a factor as many times as the power of n is;

And a m is taken as a factor as many times as the power of m is;

Therefore, degrees with the same stems can be multiplied by adding the exponents.

So, a 2 .a 6 \u003d a 2 + 6 \u003d a 8. And x 3 .x 2 .x \u003d x 3 + 2 + 1 \u003d x 6.

Or:
4a n ⋅ 2a n \u003d 8a 2n
b 2 y 3 ⋅ b 4 y \u003d b 6 y 4
(b + h - y) n ⋅ (b + h - y) \u003d (b + h - y) n + 1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are - negative.

1. So, a -2 .a -3 \u003d a -5. This can be written as (1 / aa). (1 / aaa) \u003d 1 / aaaaa.

2.y -n .y -m \u003d y -n-m.

3.a -n .a m \u003d a m-n.

If a + b is multiplied by a - b, the result is a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.

If the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degree.

So, (a - y). (A + y) \u003d a 2 - y 2.
(a 2 - y 2) ⋅ (a 2 + y 2) \u003d a 4 - y 4.
(a 4 - y 4) ⋅ (a 4 + y 4) \u003d a 8 - y 8.

Division of degrees

Power numbers can be divided, like other numbers, by subtracting from the divisor, or by placing them in fractional form.

So a 3 b 2 divided by b 2 equals a 3.

A 5 divided by a 3 looks like $ \\ frac $. But this is equal to a 2. In a series of numbers
a +4, a +3, a +2, a +1, a 0, a -1, a -2, a -3, a -4.
any number can be divided by another, and the exponent will be equal to difference exponents of divisible numbers.

When dividing degrees with the same base, their indicators are subtracted..

So, y 3: y 2 \u003d y 3-2 \u003d y 1. That is, $ \\ frac \u003d y $.

And a n + 1: a \u003d a n + 1-1 \u003d a n. That is, $ \\ frac \u003d a ^ n $.

Or:
y 2m: y m \u003d y m
8a n + m: 4a m \u003d 2a n
12 (b + y) n: 3 (b + y) 3 \u003d 4 (b + y) n-3

The rule is also true for numbers with negative values \u200b\u200bof degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $ \\ frac: \\ frac \u003d \\ frac. \\ Frac \u003d \\ frac \u003d \\ frac $.

h 2: h -1 \u003d h 2 + 1 \u003d h 3 or $ h ^ 2: \\ frac \u003d h ^ 2. \\ frac \u003d h ^ 3 $

It is necessary to master very well the multiplication and division of powers, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Decrease exponents in $ \\ frac $ Answer: $ \\ frac $.

2. Decrease exponents in $ \\ frac $. Answer: $ \\ frac $ or 2x.

3. Decrease the exponents a 2 / a 3 and a -3 / a -4 and bring them to the common denominator.
a 2 .a -4 is a -2 first numerator.
a 3 .a -3 is a 0 \u003d 1, the second numerator.
a 3 .a -4 is a -1, the common numerator.
After simplification: a -2 / a -1 and 1 / a -1.

4. Decrease the exponents 2a 4 / 5a 3 and 2 / a 4 and bring them to the common denominator.
Answer: 2a 3 / 5a 7 and 5a 5 / 5a 7 or 2a 3 / 5a 2 and 5 / 5a 2.

5. Multiply (a 3 + b) / b 4 by (a - b) / 3.

6. Multiply (a 5 + 1) / x 2 by (b 2 - 1) / (x + a).

7. Multiply b 4 / a -2 by h -3 / x and a n / y -3.

8. Divide a 4 / y 3 by a 3 / y 2. Answer: a / y.

Degree properties

We remind you that this lesson understands power properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in the lessons for grade 8.

A natural exponent has several important properties that make it easier to calculate in exponent examples.

Property number 1
Product of degrees

When multiplying degrees with the same bases, the base remains unchanged, and the exponents are added.

a m · a n \u003d a m + n, where "a" is any number, and "m", "n" are any natural numbers.

This property of degrees also affects the product of three or more degrees.

Simplify the expression.
b b 2 b 3 b 4 b 5 \u003d b 1 + 2 + 3 + 4 + 5 \u003d b 15

Present as a degree.
6 15 36 \u003d 6 15 6 2 \u003d 6 15 6 2 \u003d 6 17

Present as a degree.
(0.8) 3 (0.8) 12 \u003d (0.8) 3 + 12 \u003d (0.8) 15

Please note that in the specified property it was only about the multiplication of powers with the same bases ... It does not apply to their addition.

You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if
count (3 3 + 3 2) \u003d (27 + 9) \u003d 36, and 3 5 \u003d 243

Property number 2
Private degrees

When dividing degrees with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

Write the quotient as a degree
(2b) 5: (2b) 3 \u003d (2b) 5 - 3 \u003d (2b) 2

Calculate.

11 3 - 2 4 2 - 1 \u003d 11 4 \u003d 44
Example. Solve the equation. We use the property of private degrees.
3 8: t \u003d 3 4

Answer: t \u003d 3 4 \u003d 81

Using properties # 1 and # 2, you can easily simplify expressions and perform calculations.

Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 \u003d 4 5m + 6 + m + 2: 4 4m + 3 \u003d 4 6m + 8 - 4m - 3 \u003d 4 2m + 5

Example. Find the value of an expression using the properties of the degree.

2 11 − 5 = 2 6 = 64

Note that property 2 was only about dividing degrees with the same bases.

The difference (4 3 −4 2) cannot be replaced with 4 1. This is understandable if we calculate (4 3 −4 2) \u003d (64 - 16) \u003d 48, and 4 1 \u003d 4

Property number 3
Exponentiation

When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.

(a n) m \u003d a n · m, where "a" is any number, and "m", "n" are any natural numbers.

Note that property # 4, like other degree properties, is applied in reverse order.

(a n b n) \u003d (a b) n

That is, in order to multiply degrees with the same indicators, you can multiply the bases, and the exponent can be left unchanged.

Example. Calculate.
2 4 5 4 \u003d (2 5) 4 \u003d 10 4 \u003d 10,000

Example. Calculate.
0.5 16 2 16 \u003d (0.5 2) 16 \u003d 1

In more complex examples, there may be cases when multiplication and division must be performed over degrees with different bases and different exponents. In this case, we advise you to proceed as follows.

For example, 4 5 3 2 \u003d 4 3 4 2 3 2 \u003d 4 3 (4 3) 2 \u003d 64 12 2 \u003d 64 144 \u003d 9216

An example of raising to a decimal power.

4 21 (−0.25) 20 \u003d 4 4 20 (−0.25) 20 \u003d 4 (4 (−0.25)) 20 \u003d 4 (−1) 20 \u003d 4 1 \u003d 4

Properties 5
Degree of quotient (fraction)

To raise a quotient to a power, you can raise a separate dividend and a divisor to this power, and divide the first result by the second.

(a: b) n \u003d a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n is any natural number.

Example. Present the expression in the form of private degrees.
(5: 3) 12 = 5 12: 3 12

We remind you that the quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.

Degrees and roots

Operations with powers and roots. Degree with negative ,

zero and fractional indicator. About expressions that don't make sense.

Operations with degrees.

1. When multiplying degrees with the same base, their indicators are added:

a m · a n \u003d a m + n.

2. When dividing degrees with the same base, their indicators deducted .

3. The degree of the product of two or more factors is equal to the product of the degrees of these factors.

4. The degree of the ratio (fraction) is equal to the ratio of the degrees of the dividend (numerator) and divisor (denominator):

(a / b) n \u003d a n / b n.

5. When raising a degree to a degree, their indicators are multiplied:

All of the above formulas are read and executed in both directions from left to right and vice versa.

PRI me r. (2 · 3 · 5/15) ² = 2 ² 3 ² 5 ² / 15 ² \u003d 900/225 \u003d 4 .

Operations with roots. In all the formulas below, the symbol means arithmetic root (the radical expression is positive).

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of the ratio is equal to the ratio of the roots of the dividend and the divisor:

3. When raising a root to a power, it is enough to raise to this power root number:

4. If we increase the degree of the root by m times and at the same time raise the radical number to the m-th power, then the value of the root will not change:

5. If we reduce the degree of the root by m times and at the same time extract the mth root from the radical number, then the value of the root will not change:

Expansion of the concept of degree. Until now, we have considered degrees only with a natural exponent; but actions with powers and roots can also lead to negative, zero and fractional indicators. All these degree indicators require additional definition.

Degree with negative exponent. The power of a number with a negative (integer) exponent is defined as a unit divided by the power of the same number with an exponent equal to the absolute value of a negative exponent:

Now the formula a m : a n = a m - n can be used not only for m greater than n , but also at m less than n .

PRI me r. a 4: a 7 \u003d a 4 — 7 \u003d a — 3 .

If we want the formula a m : a n = a m — n was fair when m \u003d n , we need a definition of the zero degree.

Zero degree. The power of any nonzero number with exponent zero is 1.

EXAMPLES 2 0 \u003d 1, ( – 5) 0 = 1, (– 3 / 5) 0 = 1.

Fractional exponent. In order to raise a real number a to the power of m / n, you need to extract the nth root of the mth power of this number a:

About expressions that don't make sense. There are several such expressions.

where a ≠ 0 , does not exist.

Indeed, assuming that x - some number, then in accordance with the definition of the division operation we have: a = 0· x, i.e. a \u003d 0, which contradicts the condition: a ≠ 0

— any number.

Indeed, if we assume that this expression is equal to some number x, then according to the definition of the division operation we have: 0 \u003d 0 x ... But this equality holds for any number x, as required to prove.

0 0 — any number.

Solution. Consider three main cases:

1) x = 0 – this value does not satisfy this equation

2) at x \u003e 0 we get: x / x \u003d 1, i.e. 1 \u003d 1, whence it follows

what x - any number; but taking into account that in

our case x \u003e 0, the answer is x > 0 ;

Rules for multiplying degrees with different bases

DEGREE WITH A RATIONAL INDICATOR,

DEGREE FUNCTION IV

§ 69. Multiplication and division of powers with the same bases

Theorem 1. To multiply degrees with the same bases, it is enough to add the exponents, and leave the base the same, that is

Evidence. By definition of the degree

2 2 2 3 = 2 5 = 32; (-3) (-3) 3 = (-3) 4 = 81.

We have considered the product of two degrees. In fact, the proved property is true for any number of degrees with the same bases.

Theorem 2. To divide powers with the same bases, when the index of the dividend is greater than the index of the divisor, it is enough to subtract the index of the divisor from the index of the dividend, and leave the base the same, that is at m\u003e n

(a =/= 0)

Evidence. Recall that the quotient of dividing one number by another is a number that, when multiplied by a divisor, gives the dividend. Therefore, prove the formula where a \u003d / \u003d 0, it's like proving the formula

If a m\u003e n , then the number t - n will be natural; therefore, by Theorem 1

Theorem 2 is proved.

It should be noted that the formula

proved by us only under the assumption that m\u003e n ... Therefore, from what has been proven, one cannot draw, for example, the following conclusions:

In addition, we have not yet considered degrees with negative indicators, and we do not yet know what meaning can be given to the expression 3 - 2 .

Theorem 3. To raise a power to a power, it is enough to multiply the indicators, leaving the base of the power the same, i.e

Evidence. Using the definition of the degree and Theorem 1 of this section, we obtain:

q.E.D.

For example, (2 3) 2 \u003d 2 6 \u003d 64;

518 (Orally.) Define x from equations:

1) 2 2 2 2 3 2 4 2 5 2 6 = 2 x ; 3) 4 2 4 4 4 6 4 8 4 10 = 2 x ;

2) 3 3 3 3 5 3 7 3 9 = 3 x ; 4) 1 / 5 1 / 25 1 / 125 1 / 625 = 1 / 5 x .

519. (U st n about.) To simplify:

520. Simplify:

521. These expressions should be presented in the form of degrees with the same bases:

1) 32 and 64; 3) 8 5 and 16 3; 5) 4 100 and 32 50;

2) -1000 and 100; 4) -27 and -243; 6) 81 75 8 200 and 3 600 4 150.