Students always ask: “Why can't I use a calculator on a math exam? How to extract the square root of a number without a calculator? Let's try to answer this question.
How to extract the square root of a number without the help of a calculator?
Action square root extraction the opposite of squaring.
√81= 9 9 2 =81
If we take the square root of a positive number and square the result, we get the same number.
From small numbers that are exact squares of natural numbers, for example 1, 4, 9, 16, 25, ..., 100, square roots can be extracted verbally. Usually at school they teach a table of squares of natural numbers up to twenty. Knowing this table, it is easy to extract the square roots from the numbers 121,144, 169, 196, 225, 256, 289, 324, 361, 400. From numbers greater than 400, you can extract using the selection method using some tips. Let's try an example to consider this method.
Example: Extract the root of the number 676.
We notice that 20 2 \u003d 400, and 30 2 \u003d 900, which means 20< √676 < 900.
Exact squares of natural numbers end in 0; 1; 4; 5; 6; 9.
The number 6 is given by 4 2 and 6 2 .
So, if the root is taken from 676, then it is either 24 or 26.
It remains to check: 24 2 = 576, 26 2 = 676.
Answer: √676 = 26 .
More example: √6889 .
Since 80 2 \u003d 6400, and 90 2 \u003d 8100, then 80< √6889 < 90.
The number 9 is given by 3 2 and 7 2, then √6889 is either 83 or 87.
Check: 83 2 = 6889.
Answer: √6889 = 83 .
If you find it difficult to solve by the selection method, then you can factorize the root expression.
For example, find √893025.
Let's factorize the number 893025, remember, you did it in the sixth grade.
We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.
More example: √20736. Let's factorize the number 20736:
We get √20736 = √2 8 ∙3 4 = 2 4 ∙3 2 = 144.
Of course, factoring requires knowledge of divisibility criteria and factoring skills.
And finally, there is square root rule. Let's look at this rule with an example.
Calculate √279841.
To extract the root of a multi-digit integer, we split it from right to left into faces containing 2 digits each (there may be one digit in the left extreme face). Write like this 27'98'41
To get the first digit of the root (5), we extract the square root of the largest exact square contained in the first left face (27).
Then the square of the first digit of the root (25) is subtracted from the first face and the next face (98) is attributed (demolished) to the difference.
To the left of the resulting number 298, they write the double digit of the root (10), divide by it the number of all tens of the previously obtained number (29/2 ≈ 2), experience the quotient (102 ∙ 2 = 204 should not be more than 298) and write (2) after the first digit of the root.
Then the resulting quotient 204 is subtracted from 298, and the next facet (41) is attributed (demolished) to the difference (94).
To the left of the resulting number 9441, they write the double product of the digits of the root (52 ∙ 2 = 104), divide by this product the number of all tens of the number 9441 (944/104 ≈ 9), experience the quotient (1049 ∙ 9 = 9441) should be 9441 and write it down (9) after the second digit of the root.
We got the answer √279841 = 529.
Similarly extract roots of decimals. Only the radical number must be divided into faces so that the comma is between the faces.
Example. Find the value √0.00956484.
Just remember that if the decimal fraction has an odd number of decimal places, the square root is not exactly extracted from it.
So, now you have seen three ways to extract the root. Choose the one that suits you best and practice. To learn how to solve problems, you need to solve them. And if you have any questions, sign up for my lessons.
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In this section, we will consider arithmetic square roots.
In the case of a literal radical expression, we will assume that the letters contained under the root sign denote non-negative numbers.
1. The root of the product.
Let's consider such an example.
On the other hand, note that the number 2601 is the product of two factors, from which the root is easily extracted:
Take the square root of each factor and multiply these roots:
We got the same results when we took the root from the product under the root, and when we took the root from each factor separately and multiplied the results.
In many cases, the second way to find the result is easier, since you have to take the root of the smaller numbers.
Theorem 1. To extract the square root of the product, you can extract it from each factor separately and multiply the results.
We will prove the theorem for three factors, that is, we will prove the validity of the equality:
We carry out the proof by a direct verification, based on the definition arithmetic root. Let's say we need to prove the equality:
(A and B are non-negative numbers). A-priory square root, it means that
Therefore, it suffices to square the right side of the equality being proved and make sure that the root expression of the left side is obtained.
Let us apply this reasoning to the proof of equality (1). Let's square the right side; but the product is on the right side, and in order to square the product, it is enough to square each factor and multiply the results (see § 40);
It turned out a radical expression, standing on the left side. Hence, equality (1) is true.
We have proved the theorem for three factors. But the reasoning will remain the same if there are 4 and so on factors under the root. The theorem is true for any number of factors.
The result is easily found orally.
2. The root of the fraction.
Compute
Examination.
On the other side,
Let's prove the theorem.
Theorem 2. To extract the root of a fraction, you can extract the root separately from the numerator and denominator and divide the first result by the second.
It is required to prove the validity of the equality:
For the proof, we apply the method in which the previous theorem was proved.
Let's square the right side. Will have:
We got the radical expression on the left side. Hence, equality (2) is true.
So we have proved the following identities:
and formulated the corresponding rules for extracting the square root from the product and the quotient. Sometimes when performing transformations it is necessary to apply these identities, reading them "from right to left".
Rearranging the left and right sides, we rewrite the proven identities as follows:
To multiply the roots, you can multiply the radical expressions and extract the root from the product.
To separate the roots, you can divide the radical expressions and extract the root from the quotient.
3. The root of the degree.
Compute
slide 2
Lesson Objectives:
Review the definition of the arithmetic square root. Introduce and prove the square root theorem of a product. Learn to find. Test knowledge and skills through independent work.
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The square root of the product
Lesson plan: Updating knowledge. Learning new material. Fixing the formula with examples. Independent work. Summarizing. Homework assignment.
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Hello guys!
Let's repeat: 2. What is called the arithmetic square root of the number 3. At what value does the expression make sense? 1. What is the name of the expression
slide 5
Find:
1) 2) 3) 7 or or 7
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Today we will get acquainted with one of the properties of the arithmetic square root. We introduce and prove the theorem on the square root of the product, consider examples of its application. You will then be presented with tasks for self-examination. Good luck!
Slide 7
Let's try to solve
Consider the arithmetic root Find the value of the expression: So, So, the root of the product of two numbers is equal to the product of the roots of these numbers.
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The root of the product of non-negative factors is equal to the product of the roots of these factors. If then Theorem
Slide 9
The square root of the product
Proof: so they make sense. 4. Conclusion: (because the product of two non-negative numbers is non-negative) 5. So,
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We have considered the proof of the theorem on extracting the square root of a product. Let's move on to practical work. Now I will show you how this formula is applied when solving examples. Decide with me.
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Calculate the value of the square root using the root of the product theorem: Solve the examples:
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We solve examples:
2. Find the value of the expression:
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Quick score
And I guessed how you can use this formula for quick calculations. Watch and learn.
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Option 1 Option 2 I offer you examples for independent decision.
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 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 a 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.
We transform the original expression so that under the sign of the root it turns out positive number: 
. Now mixed number replace 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).
√2601 = 51, since (51) 2 = 2601.
On the other hand, note that the number 2601 is the product of two factors, from which the root is easily extracted:
We take the square root of each factor and multiply these roots:
√9 * √289 = 3 * 17 = 51.
We got the same results when we took the root from the product under the root, and when we took the root from each factor separately and multiplied the results.
In many cases, the second way to find the result is easier, since you have to take the root of the smaller numbers.
Theorem 1. To extract the square root of the product, you can extract it from each factor separately and multiply the results.
We will prove the theorem for three factors, that is, we will prove the validity of the equality:
We will carry out the proof directly by verification, based on the definition of an arithmetic root.
Let's say we need to prove the equality:
√A=B
(A and B are non-negative numbers). By the definition of square root, this means that
B2 = A.
Therefore, it suffices to square the right side of the equality being proved and make sure that the root expression of the left side is obtained.
Let us apply this reasoning to the proof of equality (1). Let's square the right side; but the product is on the right side, and to square the product, it is enough to square each factor and multiply the results (see § 40):
(√a √b √c) 2 = (√a) 2 (√b) 2 (√c) 2 = abc.
It turned out a radical expression, standing on the left side. Hence, equality (1) is true.
We have proved the theorem for three factors. But the reasoning will remain the same if there are 4 and so on factors under the root. The theorem is true for any number of factors.
Example.
The result is easily found orally.
2. The root of the fraction.
Let's prove the theorem.
Theorem 2. To extract the root of a fraction, you can extract the root separately from the numerator and denominator and divide the first result by the second.
It is required to prove the validity of the equality:
For the proof, we apply the method in which the previous theorem was proved.
Let's square the right side. Will have:
We got the radical expression on the left side. Hence, equality (2) is true.
So we have proved the following identities:
and formulated the corresponding rules for extracting the square root from the product and the quotient. Sometimes when performing transformations it is necessary to apply these identities, reading them "from right to left".
Rearranging the left and right sides, we rewrite the proven identities as follows:
To multiply the roots, you can multiply the radical expressions and extract the root from the product.
To separate the roots, you can divide the radical expressions and extract the root from the quotient.
3. Root of the degree.
In both examples, we ended up with the base of the radical expression to the power equal to the quotient of dividing the exponent by 2.
Let us prove this proposition in general form.
Theorem 3. If m is an even number, then
Briefly they say this: To take the square root of a power, just divide by 2 the exponent.(without changing the base).
For the proof, we use the method of verification by which Theorems 1 and 2 were proved.
Since m is an even number (by condition), it is an integer. We square the right side of equality (3), for which (see § 40) we multiply the exponent by 2, without changing the base
We got the radical expression on the left side. Hence, equality (3) is true.
Example. Calculate.
Computing 76 would require considerable time and effort. Theorem 3 allows us to find the result orally.
