Extracting a root is the inverse operation of exponentiation. That is, extracting the root of the number X, we get a number that, squared, will give the same number X.

Extracting the root is a fairly simple operation. A table of squares can make the extraction work easier. Because it is impossible to remember all the squares and roots by heart, and the numbers can be large.

Extracting the root from a number

extraction square root out of the number is simple. Moreover, this can be done not immediately, but gradually. For example, take the expression √256. Initially, it is difficult for an unknowing person to give an answer right away. Then we will take the steps. First, we divide by just the number 4, from which we take out the selected square as the root.

Draw: √(64 4), then it will be equivalent to 2√64. And as you know, according to the multiplication table 64 = 8 8. The answer will be 2*8=16.

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Complex root extraction

The square root cannot be calculated from negative numbers, because any number squared is a positive number!

A complex number is a number i that squared is -1. That is i2=-1.

In mathematics, there is a number that is obtained by taking the root of the number -1.

That is, it is possible to calculate the root of negative number, but this already applies to higher mathematics, not school.

Consider an example of such root extraction: √(-49)=7*√(-1)=7i.

Root calculator online

With the help of our calculator, you can calculate the extraction of a number from the square root:

Converting expressions containing the operation of extracting the root

The essence of the transformation of radical expressions is to decompose the radical number into simpler ones, from which the root can be extracted. Such as 4, 9, 25 and so on.

Let's take an example, √625. We divide the radical expression by the number 5. We get √(125 5), we repeat the operation √(25 25), but we know that 25 is 52. So the answer is 5*5=25.

But there are numbers for which the root cannot be calculated by this method and you just need to know the answer or have a table of squares at hand.

√289=√(17*17)=17

Outcome

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It's time to disassemble root extraction methods. They are based on the properties of the roots, in particular, on the equality, which is true for any non-negative number b.

Below we will consider in turn the main methods of extracting roots.

Let's start with the simplest case - extracting roots from natural numbers using a table of squares, a table of cubes, etc.

If the tables of squares, cubes, etc. is not at hand, it is logical to use the method of extracting the root, which involves decomposing the root number into simple factors.

Separately, it is worth dwelling on, which is possible for roots with odd exponents.

Finally, consider a method that allows you to sequentially find the digits of the value of the root.

Let's get started.

Using a table of squares, a table of cubes, etc.

In the simplest cases, tables of squares, cubes, etc. allow extracting roots. What are these tables?

The table of squares of integers from 0 to 99 inclusive (shown below) consists of two zones. The first zone of the table is located on a gray background; by selecting a certain row and a certain column, it allows you to make a number from 0 to 99. For example, let's select a row of 8 tens and a column of 3 units, with this we fixed the number 83. The second zone occupies the rest of the table. Each of its cells is located at the intersection of a certain row and a certain column, and contains the square of the corresponding number from 0 to 99 . At the intersection of our chosen row of 8 tens and column 3 of one, there is a cell with the number 6889, which is the square of the number 83.

Tables of cubes, tables of fourth powers of numbers from 0 to 99 and so on are similar to the table of squares, only they contain cubes, fourth powers, etc. in the second zone. corresponding numbers.

Tables of squares, cubes, fourth powers, etc. allow you to extract square roots, cube roots, fourth roots, etc. respectively from the numbers in these tables. Let us explain the principle of their application in extracting roots.

Let's say we need to extract the root of the nth degree from the number a, while the number a is contained in the table of nth degrees. According to this table, we find the number b such that a=b n . Then , therefore, the number b will be the desired root of the nth degree.

As an example, let's show how the cube root of 19683 is extracted using the cube table. We find the number 19 683 in the table of cubes, from it we find that this number is a cube of the number 27, therefore, .

It is clear that tables of n-th degrees are very convenient when extracting roots. However, they are often not at hand, and their compilation requires a certain amount of time. Moreover, it is often necessary to extract roots from numbers that are not contained in the corresponding tables. In these cases, one has to resort to other methods of extracting the roots.

Decomposition of the root number into prime factors

A fairly convenient way to extract the root from a natural number (if, of course, the root is extracted) is to decompose the root number into prime factors. His the essence is as follows: after it is quite easy to represent it as a degree with the desired indicator, which allows you to get the value of the root. Let's explain this point.

Let the root of the nth degree be extracted from a natural number a, and its value is equal to b. In this case, the equality a=b n is true. Number b as any natural number can be represented as a product of all its prime factors p 1 , p 2 , ..., p m in the form p 1 p 2 ... p m , and the root number a in this case is represented as (p 1 p 2 ... p m) n. Since the decomposition of the number into prime factors is unique, the decomposition of the root number a into prime factors will look like (p 1 ·p 2 ·…·p m) n , which makes it possible to calculate the value of the root as .

Note that if the factorization of the root number a cannot be represented in the form (p 1 ·p 2 ·…·p m) n , then the root of the nth degree from such a number a is not completely extracted.

Let's deal with this when solving examples.

Example.

Take the square root of 144 .

Solution.

If we turn to the table of squares given in the previous paragraph, it is clearly seen that 144=12 2 , from which it is clear that the square root of 144 is 12 .

But in the light of this point, we are interested in how the root is extracted by decomposing the root number 144 into prime factors. Let's take a look at this solution.

Let's decompose 144 to prime factors:

That is, 144=2 2 2 2 3 3 . Based on the resulting decomposition, the following transformations can be carried out: 144=2 2 2 2 3 3=(2 2) 2 3 2 =(2 2 3) 2 =12 2. Hence, .

Using the properties of the degree and properties of the roots, the solution could be formulated a little differently: .

Answer:

To consolidate the material, consider the solutions of two more examples.

Example.

Calculate the root value.

Solution.

The prime factorization of the root number 243 is 243=3 5 . Thus, .

Answer:

Example.

Is the value of the root an integer?

Solution.

To answer this question, let's decompose the root number into prime factors and see if it can be represented as a cube of an integer.

We have 285 768=2 3 3 6 7 2 . The resulting decomposition is not represented as a cube of an integer, since the degree of the prime factor 7 is not a multiple of three. Therefore, the cube root of 285,768 is not taken completely.

Answer:

No.

Extracting roots from fractional numbers

It's time to figure out how the root is extracted from fractional number. Let the fractional root number be written as p/q . According to the property of the root of the quotient, the following equality is true. From this equality it follows fraction root rule: The root of a fraction is equal to the quotient of dividing the root of the numerator by the root of the denominator.

Let's look at an example of extracting a root from a fraction.

Example.

What is the square root of common fraction 25/169 .

Solution.

According to the table of squares, we find that the square root of the numerator of the original fraction is 5, and the square root of the denominator is 13. Then . This completes the extraction of the root from an ordinary fraction 25/169.

Answer:

The root of a decimal fraction or a mixed number is extracted after replacing the root numbers with ordinary fractions.

Example.

Take the cube root of the decimal 474.552.

Solution.

Imagine the original decimal in the form of an ordinary fraction: 474.552=474552/1000. Then . It remains to extract the cube roots that are in the numerator and denominator of the resulting fraction. Because 474 552=2 2 2 3 3 3 13 13 13=(2 3 13) 3 =78 3 and 1 000=10 3 , then And . It remains only to complete the calculations .

Answer:

.

Extracting the root of a negative number

Separately, it is worth dwelling on extracting roots from negative numbers. When studying roots, we said that when the exponent of the root is an odd number, then a negative number can be under the sign of the root. We gave such notations the following meaning: for a negative number −a and an odd exponent of the root 2 n−1, we have . This equality gives rule for extracting odd roots from negative numbers: to extract the root from a negative number, you need to extract the root from the opposite positive number, and put a minus sign in front of the result.

Let's consider an example solution.

Example.

Find the root value.

Solution.

Let's transform the original expression so that a positive number appears under the root sign: . Now we replace the mixed number with an ordinary fraction: . We apply the rule of extracting the root from an ordinary fraction: . It remains to calculate the roots in the numerator and denominator of the resulting fraction: .

Here is a summary of the solution: .

Answer:

.

Bitwise Finding the Root Value

In the general case, under the root there is a number that, using the techniques discussed above, cannot be represented as the nth power of any number. But at the same time, there is a need to know the value of a given root, at least up to a certain sign. In this case, to extract the root, you can use an algorithm that allows you to consistently obtain a sufficient number of values ​​​​of the digits of the desired number.

The first step of this algorithm is to find out what is the most significant bit of the root value. To do this, the numbers 0, 10, 100, ... are successively raised to the power n until a number exceeding the root number is obtained. Then the number that we raised to the power of n in the previous step will indicate the corresponding high order.

For example, consider this step of the algorithm when extracting the square root of five. We take the numbers 0, 10, 100, ... and square them until we get a number greater than 5 . We have 0 2 =0<5 , 10 2 =100>5 , which means that the most significant digit will be the units digit. The value of this bit, as well as lower ones, will be found in the next steps of the root extraction algorithm.

All the following steps of the algorithm are aimed at successive refinement of the value of the root due to the fact that the values ​​of the next digits of the desired value of the root are found, starting from the highest and moving to the lowest. For example, the value of the root in the first step is 2 , in the second - 2.2 , in the third - 2.23 , and so on 2.236067977 ... . Let us describe how the values ​​of the bits are found.

Finding bits is carried out by enumeration of their possible values ​​0, 1, 2, ..., 9 . In this case, the nth powers of the corresponding numbers are calculated in parallel, and they are compared with the root number. If at some stage the value of the degree exceeds the radical number, then the value of the digit corresponding to the previous value is considered found, and the transition to the next step of the root extraction algorithm is made, if this does not happen, then the value of this digit is 9 .

Let us explain all these points using the same example of extracting the square root of five.

First, find the value of the units digit. We will iterate over the values ​​0, 1, 2, …, 9 , calculating respectively 0 2 , 1 2 , …, 9 2 until we get a value greater than the radical number 5 . All these calculations are conveniently presented in the form of a table:

So the value of the units digit is 2 (because 2 2<5 , а 2 3 >5 ). Let's move on to finding the value of the tenth place. In this case, we will square the numbers 2.0, 2.1, 2.2, ..., 2.9, comparing the obtained values ​​\u200b\u200bwith the root number 5:

Since 2.2 2<5 , а 2,3 2 >5 , then the value of the tenth place is 2 . You can proceed to finding the value of the hundredths place:

So the next value of the root of five is found, it is equal to 2.23. And so you can continue to find values ​​further: 2,236, 2,2360, 2,23606, 2,236067, … .

To consolidate the material, we will analyze the extraction of the root with an accuracy of hundredths using the considered algorithm.

First, we define the senior digit. To do this, we cube the numbers 0, 10, 100, etc. until we get a number greater than 2,151.186 . We have 0 3 =0<2 151,186 , 10 3 =1 000<2151,186 , 100 3 =1 000 000>2 151.186 , so the most significant digit is the tens digit.

Let's define its value.

Since 10 3<2 151,186 , а 20 3 >2,151.186 , then the value of the tens digit is 1 . Let's move on to units.

Thus, the value of the ones place is 2 . Let's move on to ten.

Since even 12.9 3 is less than the radical number 2 151.186 , the value of the tenth place is 9 . It remains to perform the last step of the algorithm, it will give us the value of the root with the required accuracy.

At this stage, the value of the root is found up to hundredths: .

In conclusion of this article, I would like to say that there are many other ways to extract roots. But for most tasks, those that we studied above are sufficient.

Bibliography.

• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

Instruction

Choose a radical number such a factor, the removal of which from under root valid expression - otherwise the operation will lose . For example, if under the sign root with an exponent equal to three (cube root) is worth number 128, then from under the sign can be taken out, for example, number 5. At the same time, the root number 128 will have to be divided by 5 cubed: ³√128 = 5∗³√(128/5³) = 5∗³√(128/125) = 5∗³√1.024. If the presence of a fractional number under the sign root does not contradict the conditions of the problem, it is possible in this form. If you need a simpler option, then first break the radical expression into such integer factors, the cube root of one of which will be an integer number m. For example: ³√128 = ³√(64∗2) = ³√(4³∗2) = 4∗³√2.

Use to select the factors of the root number, if it is not possible to calculate the degree of the number in your mind. This is especially true for root m with an exponent greater than two. If you have access to the Internet, then you can make calculations using calculators built into Google and Nigma search engines. For example, if you need to find the largest integer factor that can be taken out of the sign of the cubic root for the number 250, then go to the Google website and enter the query "6 ^ 3" to check if it is possible to take out from under the sign root six. The search engine will show a result equal to 216. Alas, 250 cannot be divided without a remainder by this number. Then enter the query 5^3. The result will be 125, and this allows you to split 250 into factors of 125 and 2, which means taking it out of the sign root number 5 leaving there number 2.

Sources:

• how to take it out from under the root
• The square root of the product

Take out from under root one of the factors is necessary in situations where you need to simplify a mathematical expression. There are cases when it is impossible to perform the necessary calculations using a calculator. For example, if letters of variables are used instead of numbers.

Instruction

Decompose the radical expression into simple factors. See which of the factors is repeated the same number of times, indicated in the indicators root, or more. For example, you need to take the root of the number a to the fourth power. In this case, the number can be represented as a*a*a*a = a*(a*a*a)=a*a3. indicator root in this case will correspond to factor a3. It must be taken out of the sign.

Extract the root of the resulting radicals separately, where possible. extraction root is the algebraic operation inverse to exponentiation. extraction root an arbitrary power from a number, find a number that, when raised to this arbitrary power, will result in a given number. If extraction root cannot be produced, leave the radical expression under the sign root the way it is. As a result of the above actions, you will make a removal from under sign root.

Related videos

note

Be careful when writing the radical expression as factors - an error at this stage will lead to incorrect results.

Helpful advice

When extracting roots, it is convenient to use special tables or tables of logarithmic roots - this will significantly reduce the time to find the correct solution.

Sources:

• root extraction sign in 2019

Simplification of algebraic expressions is required in many areas of mathematics, including the solution of equations of higher degrees, differentiation and integration. This uses several methods, including factorization. To apply this method, you need to find and take out a common factor behind brackets.

Instruction

Taking out the common factor for brackets- one of the most common decomposition methods. This technique is used to simplify the structure of long algebraic expressions, i.e. polynomials. The general can be a number, monomial or binomial, and to find it, the distributive property of multiplication is used.

Number. Look closely at the coefficients of each polynomial to see if they can be divided by the same number. For example, in the expression 12 z³ + 16 z² - 4, the obvious is factor 4. After the conversion, you get 4 (3 z³ + 4 z² - 1). In other words, this number is the least common integer divisor of all coefficients.

Mononomial. Determine if the same variable is in each of the terms of the polynomial. Let's assume that this is the case, now look at the coefficients, as in the previous case. Example: 9 z^4 - 6 z³ + 15 z² - 3 z.

Each element of this polynomial contains the variable z. In addition, all coefficients are multiples of 3. Therefore, the common factor will be the monomial 3 z: 3 z (3 z³ - 2 z² + 5 z - 1).

Binomial.For brackets general factor of two , a variable and a number, which is a general polynomial. Therefore, if factor-binomial is not obvious, then you need to find at least one root. Highlight the free term of the polynomial, this is the coefficient without a variable. Now apply the substitution method to the common expression of all integer divisors of the free term.

Consider: z^4 – 2 z³ + z² - 4 z + 4. Check if any of the integer divisors of 4 z^4 – 2 z³ + z² - 4 z + 4 = 0. Find z1 by simple substitution = 1 and z2 = 2, so brackets the binomials (z - 1) and (z - 2) can be taken out. In order to find the remaining expression, use sequential division into a column.

When solving various problems from the course of mathematics and physics, pupils and students are often faced with the need to extract roots of the second, third or nth degree. Of course, in the age of information technology, it will not be difficult to solve such a problem with a calculator. However, there are situations when it is impossible to use an electronic assistant.

For example, it is forbidden to bring electronics to many exams. In addition, the calculator may not be at hand. In such cases, it is useful to know at least some methods for manually calculating radicals.

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. At the same time, the rows of the table contain tens values, and the columns contain unit values. The cell at the intersection of a row and a column contains the square of a two-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 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 determine the answer from the column and row values. As an example, consider the calculation of the square root of 169.

We find a cell with this number in the table, horizontally we determine the tens - 1, vertically we find the units - 3. Answer: √169 = 13.

Similarly, you can calculate the roots of the cubic and n-th degree, 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 between 100 and 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 with its help it was impossible to find the root, you can try decompose the number under the root into prime factors. Prime factors are those that can be completely (without remainder) divided only by itself or by one. Examples would be 2, 3, 5, 7, 11, 13, etc.

Consider the calculation of the root using the example √576. Let's decompose it into simple factors. We get the following result: √576 = √(2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3) = √(2 ∙ 2 ∙ 2)² ∙ √3². Using the main property of the roots √a² = a, we get rid of the roots and squares, after which we calculate the answer: 2 ∙ 2 ∙ 2 ∙ 3 ​​= 24.

What to do if any of the factors does not have its own pair? For example, consider the calculation of √54. After factoring, we get the result in the following form: The non-removable part can be left under the root. For most problems in geometry and algebra, such an answer will be counted as the final one. But if there is a need to calculate approximate values, you can use the methods that will be discussed later.

Heron's method

What to do when you need to know at least approximately what the extracted root is (if it is impossible to get an integer value)? A quick and fairly accurate result is obtained by applying the Heron method.. Its essence lies in the use of an approximate formula:

√R = √a + (R - a) / 2√a,

where R is the number whose root is to be calculated, a is the nearest number whose root value is known.

Let's see how the method works in practice and evaluate how accurate it is. Let's calculate what √111 is equal to. The nearest number to 111, the root of which is known, is 121. Thus, R = 111, a = 121. Substitute the values ​​in 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 steps described earlier:

√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 repeated application of the formula, the error became quite insignificant.

Calculation of the root by division into a column

This method of finding the square root is a little more complicated 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 with an accuracy of 4 decimal places. Let's analyze the calculation algorithm using the example of an arbitrary number 1308.1912.

1. 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. We 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 can be without a pair. If the sign is missing on the right side of the number, then 0 should be added. In our case, we get 13 08.19 12.
2. Let's select the largest number whose square will be less than or equal to the first group of digits. In our case, this is 3. Let's write it on the top right; 3 is the first digit of the result. At the bottom right, we indicate 3 × 3 = 9; this will be needed for subsequent calculations. Subtract 9 from 13 in a column, we get the remainder 4.
3. Let's add the next pair of numbers to the remainder 4; we get 408.
4. Multiply the number on the top right by 2 and write it on the bottom right, adding _ x _ = to it. We get 6_ x _ =.
5. Instead of dashes, you need to substitute the same number, less than or equal to 408. We get 66 × 6 \u003d 396. Let's write 6 on the top right, since this is the second digit of the result. Subtract 396 from 408, we get 12.
6. Let's repeat steps 3-6. Since the numbers carried down are in the fractional part of the number, it is necessary to put a decimal point on the top right after 6. Let's write the doubled result with dashes: 72_ x _ =. A suitable number would be 1: 721 × 1 = 721. Let's write it down as an answer. Let's subtract 1219 - 721 = 498.
7. 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 signs for further calculations, two zeros must be added to the current number on the left.

As a result, we get the answer: √1308.1912 ≈ 36.1689. If you check the action with a calculator, you can make sure that all the characters were determined correctly.

Bitwise calculation of the square root value

The method is highly accurate. In addition, it is quite understandable and it 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 from the number 781. Let's consider in detail the sequence of actions.

1. Find out which digit of the square root value will be the highest. To do this, let's square 0, 10, 100, 1000, etc. and find out between which of them the root number is located. We get that 10²< 781 < 100², т. е. старшим разрядом будут десятки.
2. Let's take 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. In our case, we get 10² = 100, 20² = 400, 30² = 900. The value of the result n will be within 20< n <30.
3. Similarly to the previous step, the value of the units digit is selected. We alternately square 21.22, ..., 29: 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625, 26² = 676, 27² = 729, 28² = 784. We get that 27< n < 28.
4. 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.

Before the advent of calculators, students and teachers calculated square roots by hand. There are several ways to manually calculate the square root of a number. Some of them offer only an approximate solution, others give an exact answer.

Steps

Prime factorization

Factor the root number into factors that are square numbers. Depending on the root number, you will get an approximate or exact answer. Square numbers are numbers from which the whole square root can be taken. Factors are numbers that, when multiplied, give the original number. For example, the factors of the number 8 are 2 and 4, since 2 x 4 = 8, the numbers 25, 36, 49 are square numbers, since √25 = 5, √36 = 6, √49 = 7. Square factors are factors , which are square numbers. First, try to factorize the root number into square factors.

• For example, calculate the square root of 400 (manually). First try factoring 400 into square factors. 400 is a multiple of 100, that is, divisible by 25 - this is a square number. Dividing 400 by 25 gives you 16. The number 16 is also a square number. Thus, 400 can be factored into square factors of 25 and 16, that is, 25 x 16 = 400.
• This can be written as follows: √400 = √(25 x 16).

1. The square root of the product of some terms is equal to the product of the square roots of each term, that is, √(a x b) = √a x √b. Use this rule and take the square root of each square factor and multiply the results to find the answer.

   • In our example, take the square root of 25 and 16.
     • √(25 x 16)
     • √25 x √16
     • 5 x 4 = 20

2. If the radical number does not factor into two square factors (and it does in most cases), you will not be able to find the exact answer as an integer. But you can simplify the problem by decomposing the root number into a square factor and an ordinary factor (a number from which the whole square root cannot be taken). Then you will take the square root of the square factor and you will take the root of the ordinary factor.

   • For example, calculate the square root of the number 147. The number 147 cannot be factored into two square factors, but it can be factored into the following factors: 49 and 3. Solve the problem as follows:
     • = √(49 x 3)
     • = √49 x √3
     • = 7√3

3. If necessary, evaluate the value of the root. Now you can evaluate the value of the root (find an approximate value) by comparing it with the values ​​​​of the roots of square numbers that are closest (on both sides of the number line) to the root number. You will get the value of the root as a decimal fraction, which must be multiplied by the number behind the root sign.

   • Let's go back to our example. The root number is 3. The nearest square numbers to it are the numbers 1 (√1 = 1) and 4 (√4 = 2). Thus, the value of √3 lies between 1 and 2. Since the value of √3 is probably closer to 2 than to 1, our estimate is: √3 = 1.7. We multiply this value by the number at the root sign: 7 x 1.7 \u003d 11.9. If you do the calculations on a calculator, you get 12.13, which is pretty close to our answer.
     • This method also works with large numbers. For example, consider √35. The root number is 35. The nearest square numbers to it are the numbers 25 (√25 = 5) and 36 (√36 = 6). Thus, the value of √35 lies between 5 and 6. Since the value of √35 is much closer to 6 than it is to 5 (because 35 is only 1 less than 36), we can state that √35 is slightly less than 6. Checking with a calculator gives us the answer 5.92 - we were right.

4. Another way is to decompose the root number into prime factors. Prime factors are numbers that are only divisible by 1 and themselves. Write the prime factors in a row and find pairs of identical factors. Such factors can be taken out of the sign of the root.

   • For example, calculate the square root of 45. We decompose the root number into prime factors: 45 \u003d 9 x 5, and 9 \u003d 3 x 3. Thus, √45 \u003d √ (3 x 3 x 5). 3 can be taken out of the root sign: √45 = 3√5. Now we can estimate √5.
   • Consider another example: √88.
     • = √(2 x 44)
     • = √ (2 x 4 x 11)
     • = √ (2 x 2 x 2 x 11). You got three multiplier 2s; take a couple of them and take them out of the sign of the root.
     • = 2√(2 x 11) = 2√2 x √11. Now we can evaluate √2 and √11 and find an approximate answer.

   Calculating the square root manually

   Using column division

   1. This method involves a process similar to long division and gives an accurate answer. First, draw a vertical line dividing the sheet into two halves, and then draw a horizontal line to the right and slightly below the top edge of the sheet to the vertical line. Now divide the root number into pairs of numbers, starting with the fractional part after the decimal point. So, the number 79520789182.47897 is written as "7 95 20 78 91 82, 47 89 70".

      • For example, let's calculate the square root of the number 780.14. Draw two lines (as shown in the picture) and write the number in the top left as "7 80, 14". It is normal that the first digit from the left is an unpaired digit. The answer (the root of the given number) will be written on the top right.

   2. Given the first pair of numbers (or one number) from the left, find the largest integer n whose square is less than or equal to the pair of numbers (or one number) in question. In other words, find the square number that is closest to, but less than, the first pair of numbers (or single number) from the left, and take the square root of that square number; you will get the number n. Write the found n at the top right, and write down the square n at the bottom right.

      • In our case, the first number on the left will be the number 7. Next, 4< 7, то есть 2 2 < 7 и n = 2. Напишите 2 сверху справа - это первая цифра в искомом квадратном корне. Напишите 2×2=4 справа снизу; вам понадобится это число для последующих вычислений.

   3. Subtract the square of the number n you just found from the first pair of numbers (or one number) from the left. Write the result of the calculation under the subtrahend (the square of the number n).

      • In our example, subtract 4 from 7 to get 3.

   4. Take down the second pair of numbers and write it down next to the value obtained in the previous step. Then double the number at the top right and write the result at the bottom right with "_×_=" appended.

      • In our example, the second pair of numbers is "80". Write "80" after the 3. Then, doubling the number from the top right gives 4. Write "4_×_=" from the bottom right.

   5. Fill in the blanks on the right.

      • In our case, if instead of dashes we put the number 8, then 48 x 8 \u003d 384, which is more than 380. Therefore, 8 is too large a number, but 7 is fine. Write 7 instead of dashes and get: 47 x 7 \u003d 329. Write 7 from the top right - this is the second digit in the desired square root of the number 780.14.

   6. Subtract the resulting number from the current number on the left. Write the result from the previous step below the current number on the left, find the difference and write it below the subtracted one.

      • In our example, subtract 329 from 380, which equals 51.

   7. Repeat step 4. If the demolished pair of numbers is the fractional part of the original number, then put the separator (comma) of the integer and fractional parts in the desired square root from the top right. On the left, carry down the next pair of numbers. Double the number at the top right and write the result at the bottom right with "_×_=" appended.

      • In our example, the next pair of numbers to be demolished will be the fractional part of the number 780.14, so put the separator of the integer and fractional parts in the desired square root from the top right. Demolish 14 and write down at the bottom left. Double the top right (27) is 54, so write "54_×_=" at the bottom right.

   8. Repeat steps 5 and 6. Find the largest number in place of dashes on the right (instead of dashes you need to substitute the same number) so that the multiplication result is less than or equal to the current number on the left.

      • In our example, 549 x 9 = 4941, which is less than the current number on the left (5114). Write 9 on the top right and subtract the result of the multiplication from the current number on the left: 5114 - 4941 = 173.

   9. If you need to find more decimal places for the square root, write a pair of zeros next to the current number on the left and repeat steps 4, 5 and 6. Repeat steps until you get the accuracy of the answer you need (number of decimal places).

   Understanding the process

   To master this method, imagine the number whose square root you need to find as the area of ​​​​the square S. In this case, you will look for the length of the side L of such a square. Calculate the value of L for which L² = S.

   Enter a letter for each digit in your answer. Denote by A the first digit in the value of L (the desired square root). B will be the second digit, C the third and so on.

   Specify a letter for each pair of leading digits. Denote by S a the first pair of digits in the value S, by S b the second pair of digits, and so on.

   Explain the connection of this method with long division. As in the division operation, where each time we are only interested in one next digit of the divisible number, when calculating the square root, we work with a pair of digits in sequence (to obtain the next one digit in the square root value).

   1. Consider the first pair of digits Sa of the number S (Sa = 7 in our example) and find its square root. In this case, the first digit A of the sought value of the square root will be such a digit, the square of which is less than or equal to S a (that is, we are looking for such an A that satisfies the inequality A² ≤ Sa< (A+1)²). В нашем примере, S1 = 7, и 2² ≤ 7 < 3²; таким образом A = 2.

      • Let's say we need to divide 88962 by 7; here the first step will be similar: we consider the first digit of the divisible number 88962 (8) and select the largest number that, when multiplied by 7, gives a value less than or equal to 8. That is, we are looking for a number d for which the inequality is true: 7 × d ≤ 8< 7×(d+1). В этом случае d будет равно 1.

   2. Mentally imagine the square whose area you need to calculate. You are looking for L, that is, the length of the side of a square whose area is S. A, B, C are numbers in the number L. You can write it differently: 10A + B \u003d L (for a two-digit number) or 100A + 10B + C \u003d L (for three-digit number) and so on.

      • Let (10A+B)² = L² = S = 100A² + 2×10A×B + B². Remember that 10A+B is a number whose B stands for ones and A stands for tens. For example, if A=1 and B=2, then 10A+B equals the number 12. (10A+B)² is the area of ​​the whole square, 100A² is the area of ​​the large inner square, B² is the area of ​​the small inner square, 10A×B is the area of ​​each of the two rectangles. Adding the areas of the figures described, you will find the area of ​​the original square.