When solving various problems from a mathematics and physics course, pupils and students are often faced with the need to extract roots of the second, third or nth degree. Of course, in the century information technologies It won’t be difficult to solve this problem using a calculator. However, situations arise when it is impossible to use the electronic assistant.
For example, many exams do not allow you to bring electronics. In addition, you may not have a calculator at hand. In such cases, it is useful to know at least some methods for calculating radicals manually.
Finding square roots using a table of squares
One of the simplest ways to calculate roots is to using a special table. What is it and how to use it correctly?
Using the table, you can find the square of any number from 10 to 99. The rows of the table contain the values of tens, and the columns contain the values of units. The cell at the intersection of a row and a column contains a square double digit number. In order to calculate the square of 63, you need to find a row with a value of 6 and a column with a value of 3. At the intersection we will find a cell with the number 3969.
Since extracting the root is the inverse operation of squaring, to perform this action you must do the opposite: first find the cell with the number whose radical you want to calculate, then use the values of the column and row to determine the answer. As an example, consider the calculation square root 169.
We find a cell with this number in the table, horizontally we determine tens - 1, vertically we find units - 3. Answer: √169 = 13.
Similarly, you can calculate cube and nth roots using the appropriate tables.
The advantage of the method is its simplicity and the absence of additional calculations. The disadvantages are obvious: the method can only be used for a limited range of numbers (the number for which the root is found must be in the range from 100 to 9801). In addition, it will not work if the given number is not in the table.
Prime factorization
If the table of squares is not at hand or it turned out to be impossible to find the root with its help, you can try decompose the number under the root into prime factors . Prime factors are those that can be completely (without remainder) divisible only by themselves or by one. Examples could be 2, 3, 5, 7, 11, 13, etc.
Let's look at calculating the root using √576 as an example. Let's break it down into prime factors. We get the following result: √576 = √(2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3) = √(2 ∙ 2 ∙ 2)² ∙ √3². Using the basic property of roots √a² = a, we will get rid of roots and squares, and then calculate the answer: 2 ∙ 2 ∙ 2 ∙ 3 = 24.
What to do if any of the multipliers does not have its own pair? For example, consider the calculation of √54. After factorization, we obtain the result in the following form: √54 = √(2 ∙ 3 ∙ 3 ∙ 3) = √3² ∙ √(2 ∙ 3) = 3√6. The non-removable part can be left under the root. For most geometry and algebra problems, this answer will be counted as the final answer. But if there is a need to calculate approximate values, you can use methods that will be discussed below.
Heron's method
What to do when you need to at least approximately know what the extracted root is equal to (if it is impossible to obtain an integer value)? A quick and fairly accurate result is obtained by using the Heron method. Its essence is to use an approximate formula:
√R = √a + (R - a) / 2√a,
where R is the number whose root needs to be calculated, a is the nearest number whose root value is known.
Let's look at how the method works in practice and evaluate how accurate it is. Let's calculate what √111 is equal to. The number closest to 111, the root of which is known, is 121. Thus, R = 111, a = 121. Substitute the values into the formula:
√111 = √121 + (111 - 121) / 2 ∙ √121 = 11 - 10 / 22 ≈ 10,55.
Now let's check the accuracy of the method:
10.55² = 111.3025.
The error of the method was approximately 0.3. If the accuracy of the method needs to be improved, you can repeat the previously described steps:
√111 = √111,3025 + (111 - 111,3025) / 2 ∙ √111,3025 = 10,55 - 0,3025 / 21,1 ≈ 10,536.
Let's check the accuracy of the calculation:
10.536² = 111.0073.
After re-applying the formula, the error became completely insignificant.
Calculating the root by long division
This method of finding the square root value is a little more complex than the previous ones. However, it is the most accurate among other calculation methods without a calculator.
Let's say that you need to find the square root accurate to 4 decimal places. Let's analyze the calculation algorithm using the example of an arbitrary number 1308.1912.
- Divide the sheet of paper into 2 parts with a vertical line, and then draw another line from it to the right, slightly below the top edge. Let's write the number on the left side, dividing it into groups of 2 digits, moving to the right and left of the decimal point. The very first digit on the left may be without a pair. If the sign is missing on the right side of the number, then you should add 0. In our case, the result will be 13 08.19 12.
- Let's choose the best big number, the square of which will be less than or equal to the first group of digits. In our case it is 3. Let's write it on the top right; 3 is the first digit of the result. On the bottom right we indicate 3×3 = 9; this will be needed for subsequent calculations. From 13 in the column we subtract 9, we get a remainder of 4.
- Let's assign the next pair of numbers to remainder 4; we get 408.
- Multiply the number at the top right by 2 and write it down at the bottom right, adding _ x _ = to it. We get 6_ x _ =.
- Instead of dashes, you need to substitute the same number, less than or equal to 408. We get 66 × 6 = 396. We write 6 from the top right, since this is the second digit of the result. Subtract 396 from 408, we get 12.
- Let's repeat steps 3-6. Since the digits moved down are in the fractional part of the number, it is necessary to put decimal point on the top right after 6. Let's write down the double result with dashes: 72_ x _ =. A suitable number would be 1: 721×1 = 721. Let's write it down as the answer. Let's subtract 1219 - 721 = 498.
- Let's perform the sequence of actions given in the previous paragraph three more times to get the required number of decimal places. If there are not enough characters for further calculations, you need to add two zeros to the current number on the left.
As a result, we get the answer: √1308.1912 ≈ 36.1689. If you check the action using a calculator, you can make sure that all signs were identified correctly.
Bitwise square root calculation
The method is highly accurate. In addition, it is quite understandable and does not require memorizing formulas or a complex algorithm of actions, since the essence of the method is to select the correct result.
Let's extract the root of the number 781. Let's look at the sequence of actions in detail.
- Let's find out which digit of the square root value will be the most significant. To do this, let’s square 0, 10, 100, 1000, etc. and find out between which of them the radical number is located. We get that 10²< 781 < 100², т. е. старшим разрядом будут десятки.
- Let's choose the value of tens. To do this, we will take turns raising to the power of 10, 20, ..., 90 until we get a number greater than 781. For our case, we get 10² = 100, 20² = 400, 30² = 900. The value of the result n will be within 20< n <30.
- Similar to the previous step, the value of the units digit is selected. Let's square 21.22, ..., 29 one by one: 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625, 26² = 676, 27² = 729, 28² = 784. We get that 27< n < 28.
- Each subsequent digit (tenths, hundredths, etc.) is calculated in the same way as shown above. Calculations are carried out until the required accuracy is achieved.
Video
This video will show you how to find square roots without using a calculator.
I looked again at the sign... And, let's go!
Let's start with something simple:
Just a minute. this, which means we can write it like this:
Got it? Here's the next one for you:
Are the roots of the resulting numbers not exactly extracted? No problem - here are some examples:
What if there are not two, but more multipliers? The same! The formula for multiplying roots works with any number of factors:
Now completely on your own:
Answers: Well done! Agree, everything is very easy, the main thing is to know the multiplication table!
Root division
We've sorted out the multiplication of roots, now let's move on to the property of division.
Let me remind you that the general formula looks like this:
Which means that the root of the quotient is equal to the quotient of the roots.
Well, let's look at some examples:
That's all science is. Here's an example:
Everything is not as smooth as in the first example, but, as you can see, there is nothing complicated.
What if you come across this expression:
You just need to apply the formula in the opposite direction:
And here's an example:
You may also come across this expression:
Everything is the same, only here you need to remember how to translate fractions (if you don’t remember, look at the topic and come back!). Do you remember? Now let's decide!
I am sure that you have coped with everything, now let’s try to raise the roots to degrees.
Exponentiation
What happens if the square root is squared? It's simple, remember the meaning of the square root of a number - this is a number whose square root is equal to.
So, if we square a number whose square root is equal, what do we get?
Well, of course, !
Let's look at examples:
It's simple, right? What if the root is to a different degree? It's OK!
Follow the same logic and remember the properties and possible actions with degrees.
Read the theory on the topic “” and everything will become extremely clear to you.
For example, here is an expression:
In this example, the degree is even, but what if it is odd? Again, apply the properties of exponents and factor everything:
Everything seems clear with this, but how to extract the root of a number to a power? Here, for example, is this:
Pretty simple, right? What if the degree is greater than two? We follow the same logic using the properties of degrees:
Well, is everything clear? Then solve the examples yourself:
And here are the answers:
Entering under the sign of the root
What haven’t we learned to do with roots! All that remains is to practice entering the number under the root sign!
It's really easy!
Let's say we have a number written down
What can we do with it? Well, of course, hide the three under the root, remembering that the three is the square root of!
Why do we need this? Yes, just to expand our capabilities when solving examples:
How do you like this property of roots? Does it make life much easier? For me, that's exactly right! Only We must remember that we can only enter positive numbers under the square root sign.
Solve this example yourself -
Did you manage? Let's see what you should get:
Well done! You managed to enter the number under the root sign! Let's move on to something equally important - let's look at how to compare numbers containing a square root!
Comparison of roots
Why do we need to learn to compare numbers that contain a square root?
Very simple. Often, in large and long expressions encountered in the exam, we receive an irrational answer (remember what this is? We already talked about this today!)
We need to place the received answers on the coordinate line, for example, to determine which interval is suitable for solving the equation. And here the problem arises: there is no calculator in the exam, and without it, how can you imagine which number is greater and which is less? That's it!
For example, determine which is greater: or?
You can’t tell right away. Well, let's use the disassembled property of entering a number under the root sign?
Then go ahead:
Well, obviously, the larger the number under the root sign, the larger the root itself!
Those. if, then, .
From this we firmly conclude that. And no one will convince us otherwise!
Extracting roots from large numbers
Before this, we entered a multiplier under the sign of the root, but how to remove it? You just need to factor it into factors and extract what you extract!
It was possible to take a different path and expand into other factors:
Not bad, right? Any of these approaches is correct, decide as you wish.
Factoring is very useful when solving such non-standard problems as this:
Let's not be afraid, but act! Let's decompose each factor under the root into separate factors:
Now try it yourself (without a calculator! It won’t be on the exam):
Is this the end? Let's not stop halfway!
That's all, it's not so scary, right?
Happened? Well done, that's right!
Now try this example:
But the example is a tough nut to crack, so you can’t immediately figure out how to approach it. But, of course, we can handle it.
Well, let's start factoring? Let us immediately note that you can divide a number by (remember the signs of divisibility):
Now, try it yourself (again, without a calculator!):
Well, did it work? Well done, that's right!
Let's sum it up
- The square root (arithmetic square root) of a non-negative number is a non-negative number whose square is equal to.
. - If we simply take the square root of something, we always get one non-negative result.
- Properties of an arithmetic root:
- When comparing square roots, it is necessary to remember that the larger the number under the root sign, the larger the root itself.
How's the square root? All clear?
We tried to explain to you without any fuss everything you need to know in the exam about the square root.
It's your turn. Write to us whether this topic is difficult for you or not.
Did you learn something new or was everything already clear?
Write in the comments and good luck on your exams!
Today we will figure out on this page of our website what the square root of 100 is. Let's figure out together what the square root of 100 is, since 1000 scientists have been racking their brains on this topic for many decades, and many have come to the inevitable conclusion from calculations that such a root does not exist at all and it is simply impossible to calculate it. It is also very important in this case to ask exactly the right question to identify the square root of 100. To be precise, we will calculate the arithmetic square root of 100, since in the ordinary square root of 100 we will end up with two numbers: 10 and - 10.
We can calculate the sum of these numbers we need using a simple arithmetic technique using a vertical, familiar line, numbers and roots that are written in the lower right. There we will find the square of units of the root we need, then multiply the tens and find the double and not triple the product of the ten of any root by units. We will have to square some numbers so that the total becomes a two-digit number; if in the end we get the number 10, then we have done everything right with you. The main thing is to initially become at least a little familiar with mathematics and the mathematical progression of composing the square root before starting calculations.
Remember one single and basic rule: in order to extract the necessary square root from any integer, first of all we extract any root we need from the number of its sums and hundreds. If the number is equal to or greater than 100, then we begin to look for the root of the hundreds of actual numbers of these hundreds, then of the tens of thousands of the actual number, especially if the given number is much more than 100, then we necessarily extract the root of the number of hundreds of tens of thousands or to be more precise: out of a million of a given number. There are many rules and various scientific recommendations on this topic; school programs for extracting the square root of the number 100 will always remain unchanged.
If we consider the progress of finding the root of the number 100, we need to pay attention to the fact that there are as many digits in the root as there are under a finite number of sides, while the left side can consist of only one digit. Based on all this, the most accurate square root of any number on planet earth will be the sum of numbers whose square is exactly equal to the given number when calculated. This is where we can finish our short course on calculating the square root of 100 which will be equal to (10) ten.
Konstantinova Vera
How to find the root of a number
The problem of finding a root in mathematics is the inverse problem of raising a number to a power. There are different roots: roots of the second degree, roots of the third degree, roots of the fourth degree, and so on. It depends on what power the number was originally raised to. The root is denoted by the symbol: √ is a square root, that is, the root of the second degree; if the root has a degree greater than the second, then the corresponding degree is assigned above the root sign. The number that is under the root sign is a radical expression. When finding a root, there are several rules that will help you not make a mistake in finding the root:
- An even root (if the degree is 2, 4, 6, 8, etc.) of a negative number does NOT exist. If the radical expression is negative, but the root of an odd degree is sought (3, 5, 7, and so on), then the result will be negative.
- The root of any power of one is always one: √1 = 1.
- The root of zero is zero: √0 = 0.
How to find the root of 100
If the problem does not say what root of the degree needs to be found, then it usually means that it is necessary to find the root of the second degree (square).
Let's find √100 = ? We need to find a number that, when raised to the second power, gives the number 100. Obviously, such a number is the number 10, since: 10 2 = 100. Therefore, √100 = 10: the square root of 100 is 10.
The problem of finding a root in mathematics is the inverse problem of raising a number to a power. There are different roots: roots of the second degree, roots of the third degree, roots of the fourth degree, and so on. It depends on what power the number was originally raised to. The root is indicated by the symbol: √ is a square root, that is, the root of the second degree; if the root has a degree greater than the second, then the corresponding degree is assigned above the root sign. The number that is under the root sign is a radical expression. When finding a root, there are several rules that will help you not make a mistake in finding the root:
- An even root (if the degree is 2, 4, 6, 8, etc.) of a negative number does NOT exist. If the radical expression is negative, but the root of an odd degree is sought (3, 5, 7, and so on), then the result will be negative.
- The root of any power of one is always one: √1 = 1.
- The root of zero is zero: √0 = 0.
How to find the root of 100
If the problem does not say what root of the degree needs to be found, then it usually means that it is necessary to find the root of the second degree (square).
Let's find √100 = ? We need to find a number that, when raised to the second power, gives the number 100. Obviously, such a number is the number 10, since: 10 2 = 100. Therefore, √100 = 10: the square root of 100 is 10.
Well, if we take into account that this very square root is the product of the same number (that is, b = a), then the square root of one hundred will be 10 (100 = 10).
It should be noted that the number 100 can be represented as the product of 25 and 4. And then calculate the square root of both 25 and 4. 5 and 2. Multiply and also get 10.
When we first started studying this topic at school, square root of 100 was probably one of the easiest to understand and calculations. Usually I looked at an even (!) number of zeros and immediately calculated which number, multiplied by itself, gives the figure under the square root. For example, if it were 10000, then the square root of that number would be one hundred (100x100 = 10000). If the number under sq. the root is six zeros, then the answer will contain three zeros. Etc.
In this case, there are only two zeros in the number, which means that there were two tens. So, The square root of 100 is 10. We check: 10x10 = 100
There are several ways to calculate the square root.
1) Take a calculator or smartphone/tablet/computer with a calculation program installed, enter the number 100 and click on the square root icon, which looks something like this:
2) Know the table of squares of numbers up to 100=25*4.
3) By division method.
4) By the method of decomposition into prime factors 100=10*10.
Theoretically, if you do everything correctly, you will get a result of 10.
The icon used to represent a square root is called a radical and looks like this.
And the square root of 100 is easy to extract if you know the squares of numbers. 10 X 10 = 100. So the square root of 100, following the definition of a square root, is 10.
Probably every schoolchild knows that the number 100 is the product of 10 by 10.
Since the square root is a number that, when multiplied by itself, is a radical expression, then The square root of one hundred is equal to the number 10.
If you forgot that 100=10*10, then you can use the properties of roots:
root of 100 = root of (25*4) = root of 25 * root of 4.
Everyone knows that 5*5 = 25, and 2*2 = 4. Therefore, the root of 100 = 5 * 2 = 10.
Well, if you don’t know this, then you can use a calculator or Excel tables, they have a special formula called ROOT. Here's how it all looks visually:
Nowadays, using a calculator it is very easy to calculate the square root of any number.
You can extract the square root of 100 orally. After all, it is known that bringing the number x to the square is the number x multiplied by the number x.
If 10 10 = 100, then the square root of 100 is 10.
Answer to the question: 10 .
The square root in mathematics is denoted by a conventional symbol.
The square root of a number is a non-negative number whose square is equal to a. Since 10^2=100, the square root of 100 is 10.
There are numbers whose roots are very easy to remember. For me, this is, for example, 25 - the root will be 5, since 5*5=25, 625 is the root of 25, since 25*25=625.
I also include the number 100 as such numbers - the root will be 10, check 10*10=100. So it's correct.
Square root of one hundred? looks like it will be 10
It’s hard to imagine that a person would go online to find this answer, but if we imagine that he is completely uncollected and inattentive, then I give the answer. The square root of the number 100 is 10, and also -10. Many sources write it this way.
The square root of 100 has two values: 10 and -10. Those who don’t believe can check by multiplying.
To extract the square root without a calculator, you need to resort to decomposing the number under the root into the smallest factors and proceed from there. So for the number one hundred:
And accordingly, from here it immediately becomes clear that the square root of one hundred will be exactly 10.
I had to remember a rule that I remembered from school:
Although extracting the root of 100 is a simple matter that does not require the use of calculators, since it is ingrained in memory for life. The number 100 is obtained by multiplying 10 by 10, and therefore the number 10 and will be the root of a hundred.