Sinusitis

How to extract the root of the number 8. Square root

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

Steps

Prime factorization

Factor the radical number into factors that are square numbers. Depending on the radical 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 factor the radical number into square factors.

For example, calculate the square root of 400 (by hand). 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 the square factors of 25 and 16, that is, 25 x 16 = 400.
This can be written as follows: √400 = √(25 x 16).

The square root of the product of some terms is equal to the product square roots from each term, that is, √(a x b) = √a x √b. Use this rule to take the square root of each square factor and multiply the results to find the answer.
- In our example, take the root of 25 and 16.
  - √(25 x 16)
  - √25 x √16
  - 5 x 4 = 20
If the radical number does not factor into two square factors (and this happens in most cases), you will not be able to find the exact answer in the form of a whole number. But you can simplify the problem by decomposing the radical 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 will take the root of the common 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 factorized into the following factors: 49 and 3. Solve the problem as follows:
  - = √(49 x 3)
  - = √49 x √3
  - = 7√3
If necessary, estimate the value of the root. Now you can estimate the value of the root (find an approximate value) by comparing it with the values of the roots of the square numbers that are closest (on both sides of the number line) to the radical number. You will get the value of the root as decimal, which must be multiplied by the number behind the root sign.
- Let's return to our example. The radical number is 3. The square numbers closest to it will be the numbers 1 (√1 = 1) and 4 (√4 = 2). Thus, the value of √3 is located 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 = 11.9. If you do the math on a calculator, you'll get 12.13, which is pretty close to our answer.
  - This method also works with large numbers. For example, consider √35. The radical number is 35. The closest square numbers to it will be the numbers 25 (√25 = 5) and 36 (√36 = 6). Thus, the value of √35 is located between 5 and 6. Since the value of √35 is much closer to 6 than to 5 (because 35 is only 1 less than 36), we can say that √35 is slightly less than 6. Check on the calculator gives us the answer 5.92 - we were right.
Another way is to factor the radical number into prime factors. Prime factors are numbers that are divisible only by 1 and themselves. Write it down prime factors in a row and find pairs of identical factors. Such factors can be taken out of the root sign.
- For example, calculate the square root of 45. We factor the radical number into prime factors: 45 = 9 x 5, and 9 = 3 x 3. Thus, √45 = √(3 x 3 x 5). 3 can be taken out as a root sign: √45 = 3√5. Now we can estimate √5.
- Let's look at another example: √88.
  - = √(2 x 44)
  - = √ (2 x 4 x 11)
  - = √ (2 x 2 x 2 x 11). You received three multipliers of 2; take a couple of them and move them beyond the root sign.
  - = 2√(2 x 11) = 2√2 x √11. Now you can evaluate √2 and √11 and find an approximate answer.
Calculating square root manually

Using long division
1. This method involves a process similar to long division and provides an accurate answer. First, draw a vertical line dividing the sheet into two halves, and then to the right and slightly below the top edge of the sheet, draw a horizontal line to the vertical line. Now divide the radical 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 given number in the form “7 80, 14” at the top left. It is normal that the first digit from the left is an unpaired digit. You will write the answer (the root of this number) at the top right.
2. For the first pair of numbers (or single number) from the left, find the largest integer n whose square is less than or equal to the pair of numbers (or single number) in question. In other words, find the square number that is closest to, but smaller 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 n you found at the top right, and write the square of n at the bottom right.
  - In our case, the first number on the left will be 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 single number) on the left. Write the result of the calculation under the subtrahend (the square of the number n).
  - In our example, subtract 4 from 7 and 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 the addition of "_×_=".
  - In our example, the second pair of numbers is "80". Write "80" after the 3. Then, double the number on the top right gives 4. Write "4_×_=" on the bottom right.
5. Fill in the blanks on the right.
  - In our case, if we put the number 8 instead of dashes, then 48 x 8 = 384, which is more than 380. Therefore, 8 is too large a number, but 7 will do. Write 7 instead of dashes and get: 47 x 7 = 329. Write 7 at 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 under the current number on the left, find the difference and write it under the subtrahend.
  - In our example, subtract 329 from 380, which equals 51.
7. Repeat step 4. If the pair of numbers being carried is the fractional part original number, then put a separator (comma) between the integer and fractional parts in the desired square root at the top right. On the left, bring down the next pair of numbers. Double the number at the top right and write the result at the bottom right with the addition of "_×_=".
  - In our example, the next pair of numbers to be removed will be the fractional part of the number 780.14, so place the separator of the integer and fractional parts in the desired square root in the upper right. Take down 14 and write it in the bottom left. Double the number on the top right (27) is 54, so write "54_×_=" on the bottom right.
8. Repeat steps 5 and 6. Find one greatest number in place of the dashes on the right (instead of the dashes you need to substitute the same number) so that the result of the multiplication 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 couple of zeros to the left of the current number and repeat steps 4, 5, and 6. Repeat steps until you get the answer precision (number of decimal places) you need.
Understanding the Process
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 desired square root value will be a digit whose square is less than or equal to S a (that is, we are looking for an A such that 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 a 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 equal to S. A, B, C are the numbers in the number L. You can write it differently: 10A + B = L (for a two-digit number) or 100A + 10B + C = 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 in which the digit B stands for units and the digit A stands for tens. For example, if A=1 and B=2, then 10A+B is equal to the number 12. (10A+B)² is the area of the entire square, 100A²- area of the large inner square, B²- area of the small inner square, 10A×B- the area of each of the two rectangles. By adding up the areas of the described figures, you will find the area of the original square.

Mathematics originated when man became aware of himself and began to position himself as an autonomous unit of the world. The desire to measure, compare, count what surrounds you is what underlay one of the fundamental sciences of our days. At first, these were particles of elementary mathematics, which made it possible to connect numbers with their physical expressions, later the conclusions began to be presented only theoretically (due to their abstraction), but after a while, as one scientist put it, “mathematics reached the ceiling of complexity when they disappeared from it.” all the numbers." The concept of “square root” appeared at a time when it could be easily supported by empirical data, going beyond the plane of calculations.

Where it all began

The first mention of the root, which is at the moment denoted as √, was recorded in the works of Babylonian mathematicians, who laid the foundation for modern arithmetic. Of course, they bore little resemblance to the current form - scientists of those years first used bulky tablets. But in the second millennium BC. e. They derived an approximate calculation formula that showed how to extract the square root. The photo below shows a stone on which Babylonian scientists carved the process for deducing √2, and it turned out to be so correct that the discrepancy in the answer was found only in the tenth decimal place.

In addition, the root was used if it was necessary to find a side of a triangle, provided that the other two were known. Well, when solving quadratic equations, there is no escape from extracting the root.

Along with the Babylonian works, the object of the article was also studied in the Chinese work “Mathematics in Nine Books,” and the ancient Greeks came to the conclusion that any number from which the root cannot be extracted without a remainder gives an irrational result.

The origin of this term is associated with the Arabic representation of number: ancient scientists believed that the square of an arbitrary number grows from a root, like a plant. In Latin, this word sounds like radix (you can trace a pattern - everything that has a “root” meaning is consonant, be it radish or radiculitis).

Scientists of subsequent generations picked up this idea, designating it as Rx. For example, in the 15th century, in order to indicate that the square root of an arbitrary number a was taken, they wrote R 2 a. Habitual modern view"tick" √ appeared only in the 17th century thanks to Rene Descartes.

Our days

In mathematical terms, the square root of a number y is the number z whose square is equal to y. In other words, z 2 =y is equivalent to √y=z. However this definition relevant only for arithmetic root, since it implies a non-negative value of the expression. In other words, √y=z, where z is greater than or equal to 0.

In general, which applies to determining an algebraic root, the value of the expression can be either positive or negative. Thus, due to the fact that z 2 =y and (-z) 2 =y, we have: √y=±z or √y=|z|.

Due to the fact that the love for mathematics has only increased with the development of science, there are various manifestations of affection for it that are not expressed in dry calculations. For example, along with such interesting phenomena as Pi Day, square root holidays are also celebrated. They are celebrated nine times every hundred years, and are determined by to the following principle: numbers that indicate in order the day and month, must be the square root of the year. So, the next time we will celebrate this holiday is April 4, 2016.

Properties of the square root on the field R

Almost everything mathematical expressions have a geometric basis, this fate did not escape √y, which is defined as the side of a square with area y.

How to find the root of a number?

There are several calculation algorithms. The simplest, but at the same time quite cumbersome, is the usual arithmetic calculation, which is as follows:

1) from the number whose root we need, odd numbers are subtracted in turn - until the remainder at the output is less than the subtracted one or even equal to zero. The number of moves will ultimately become the desired number. For example, calculating the square root of 25:

The next odd number is 11, the remainder is: 1<11. Количество ходов - 5, так что корень из 25 равен 5. Вроде все легко и просто, но представьте, что придется вычислять из 18769?

For such cases there is a Taylor series expansion:

√(1+y)=∑((-1) n (2n)!/(1-2n)(n!) 2 (4 n))y n , where n takes values from 0 to

+∞, and |y|≤1.

Graphic representation of the function z=√y

Consider the elementary function z=√y on the field of real numbers R, where y is greater than or equal to zero. Its schedule looks like this:

The curve grows from the origin and necessarily intersects the point (1; 1).

Properties of the function z=√y on the field of real numbers R

1. The domain of definition of the function under consideration is the interval from zero to plus infinity (zero is included).

2. The range of values of the function under consideration is the interval from zero to plus infinity (zero is again included).

3. The function takes its minimum value (0) only at the point (0; 0). There is no maximum value.

4. The function z=√y is neither even nor odd.

5. The function z=√y is not periodic.

6. There is only one point of intersection of the graph of the function z=√y with the coordinate axes: (0; 0).

7. The intersection point of the graph of the function z=√y is also the zero of this function.

8. The function z=√y is continuously growing.

9. The function z=√y takes only positive values, therefore, its graph occupies the first coordinate angle.

Options for displaying the function z=√y

In mathematics, to facilitate the calculation of complex expressions, the power form of writing the square root is sometimes used: √y=y 1/2. This option is convenient, for example, in raising a function to a power: (√y) 4 =(y 1/2) 4 =y 2. This method is also a good representation for differentiation with integration, since thanks to it the square root is represented as an ordinary power function.

And in programming, replacing the symbol √ is the combination of letters sqrt.

It is worth noting that in this area the square root is in great demand, since it is part of most geometric formulas necessary for calculations. The counting algorithm itself is quite complex and is based on recursion (a function that calls itself).

Square root in complex field C

By and large, it was the subject of this article that stimulated the discovery of the field of complex numbers C, since mathematicians were haunted by the question of obtaining an even root of a negative number. This is how the imaginary unit i appeared, which is characterized by a very interesting property: its square is -1. Thanks to this, quadratic equations were solved even with a negative discriminant. In C, the same properties are relevant for the square root as in R, the only thing is that the restrictions on the radical expression are removed.

Quite often, when solving problems, we are faced with large numbers from which we need to extract square root. Many students decide that this is a mistake and begin to re-solve the entire example. Under no circumstances should you do this! There are two reasons for this:

Roots of large numbers do appear in problems. Especially in text ones;
There is an algorithm by which these roots are calculated almost orally.

We will consider this algorithm today. Perhaps some things will seem incomprehensible to you. But if you pay attention to this lesson, you will receive a powerful weapon against square roots.

So, the algorithm:

Limit the required root above and below to numbers that are multiples of 10. Thus, we will reduce the search range to 10 numbers;
From these 10 numbers, weed out those that definitely cannot be roots. As a result, 1-2 numbers will remain;
Square these 1-2 numbers. The one whose square is equal to the original number will be the root.

Before putting this algorithm into practice, let's look at each individual step.

Root limitation

First of all, we need to find out between which numbers our root is located. It is highly desirable that the numbers be multiples of ten:

10 2 = 100;
20 2 = 400;
30 2 = 900;
40 2 = 1600;
...
90 2 = 8100;
100 2 = 10 000.

We get a series of numbers:

100; 400; 900; 1600; 2500; 3600; 4900; 6400; 8100; 10 000.

What do these numbers tell us? It's simple: we get boundaries. Take, for example, the number 1296. It lies between 900 and 1600. Therefore, its root cannot be less than 30 and greater than 40:

[Caption for the picture]

The same thing applies to any other number from which you can find the square root. For example, 3364:

[Caption for the picture]

Thus, instead of an incomprehensible number, we get a very specific range in which the original root lies. To further narrow the search area, move on to the second step.

Eliminating obviously unnecessary numbers

So, we have 10 numbers - candidates for the root. We got them very quickly, without complex thinking and multiplication in a column. It's time to move on.

Believe it or not, we'll now reduce the number of candidate numbers to two - again without any complicated calculations! It is enough to know the special rule. Here it is:

The last digit of the square depends only on the last digit original number.

In other words, just look at the last digit of the square and we will immediately understand where the original number ends.

There are only 10 digits that can come in last place. Let's try to find out what they turn into when squared. Take a look at the table:

1	2	3	4	5	6	7	8	9	0
1	4	9	6	5	6	9	4	1	0

This table is another step towards calculating the root. As you can see, the numbers in the second line turned out to be symmetrical with respect to the five. For example:

2 2 = 4;
8 2 = 64 → 4.

As you can see, the last digit is the same in both cases. This means that, for example, the root of 3364 necessarily ends in 2 or 8. On the other hand, we remember the restriction from the previous paragraph. We get:

[Caption for the picture]

Red squares indicate that we do not yet know this figure. But the root lies in the range from 50 to 60, on which there are only two numbers ending in 2 and 8:

[Caption for the picture]

That's it! Of all the possible roots, we left only two options! And this is in the most difficult case, because the last digit can be 5 or 0. And then there will be only one candidate for the roots!

Final calculations

So, we have 2 candidate numbers left. How do you know which one is the root? The answer is obvious: square both numbers. The one that squared gives the original number will be the root.

For example, for the number 3364 we found two candidate numbers: 52 and 58. Let's square them:

52 2 = (50 +2) 2 = 2500 + 2 50 2 + 4 = 2704;
58 2 = (60 − 2) 2 = 3600 − 2 60 2 + 4 = 3364.

That's it! It turned out that the root is 58! At the same time, to simplify the calculations, I used the formula for the squares of the sum and difference. Thanks to this, I didn’t even have to multiply the numbers into a column! This is another level of optimization of calculations, but, of course, it is completely optional :)

Examples of calculating roots

Theory is, of course, good. But let's check it in practice.

[Caption for the picture]

First, let's find out between which numbers the number 576 lies:

400 < 576 < 900
20 2 < 576 < 30 2

Now let's look at the last number. It is equal to 6. When does this happen? Only if the root ends in 4 or 6. We get two numbers:

All that remains is to square each number and compare it with the original:

24 2 = (20 + 4) 2 = 576

Great! The first square turned out to be equal to the original number. So this is the root.

Task. Calculate the square root:
[Caption for the picture]

900 < 1369 < 1600;
30 2 < 1369 < 40 2;

Let's look at the last digit:

1369 → 9;
33; 37.

Square it:

33 2 = (30 + 3) 2 = 900 + 2 30 3 + 9 = 1089 ≠ 1369;
37 2 = (40 − 3) 2 = 1600 − 2 40 3 + 9 = 1369.

Here is the answer: 37.

Task. Calculate the square root:
[Caption for the picture]

We limit the number:

2500 < 2704 < 3600;
50 2 < 2704 < 60 2;

Let's look at the last digit:

2704 → 4;
52; 58.

Square it:

52 2 = (50 + 2) 2 = 2500 + 2 50 2 + 4 = 2704;

We received the answer: 52. The second number will no longer need to be squared.

Task. Calculate the square root:
[Caption for the picture]

We limit the number:

3600 < 4225 < 4900;
60 2 < 4225 < 70 2;

Let's look at the last digit:

4225 → 5;
65.

As you can see, after the second step there is only one option left: 65. This is the desired root. But let’s still square it and check:

65 2 = (60 + 5) 2 = 3600 + 2 60 5 + 25 = 4225;

Everything is correct. We write down the answer.

Conclusion

Alas, no better. Let's look at the reasons. There are two of them:

In any normal mathematics exam, be it the State Examination or the Unified State Exam, the use of calculators is prohibited. And if you bring a calculator into class, you can easily be kicked out of the exam.
Don't be like stupid Americans. Which are not like roots - they cannot add two prime numbers. And when they see fractions, they generally become hysterical.

Instructions

Select a multiplier for the radical number, the removal of which from under root is really an expression - otherwise the operation will lose . For example, if under the sign root with an exponent equal to three (cube root), it costs number 128, then from under the sign you can take out, for example, number 5. At the same time, the radical 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, then 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 factors of a radical number if it is not possible to calculate the powers of a number in your head. This is especially true for root m with an exponent greater than two. If you have access to the Internet, you can perform calculations using the calculators built into the Google and Nigma search engines. For example, if you need to find the largest integer factor that can be taken out from under the cubic sign root for the number 250, then go to the Google website and enter the query “6^3” to check if it is possible to remove it 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 divide 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 get it out from under the roots
Square root of the product

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

Instructions

Break down 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 fourth root of a. In this case, the number can be represented as a*a*a*a = a*(a*a*a)=a*a3. Indicator root in this case it will correspond with factor a3. It needs to 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 of an arbitrary power, find a number from a number that, when raised to this arbitrary power, will result in the given number. If extraction root cannot be produced, leave the radical expression under the sign root just the way it is. As a result of the above actions, you will be removed from under sign root.

Video on the topic

Please note

Be careful when writing radical expressions in the form of factors - an error at this stage will lead to incorrect results.

Useful advice

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

Sources:

root extraction sign in 2019

Simplification of algebraic expressions is required in many areas of mathematics, including solving higher-order equations, differentiation and integration. Several methods are used, including factorization. To apply this method, you need to find and make a general factor for brackets.

Instructions

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

Number. Look carefully 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 it is obvious factor 4. After the transformation, you get 4 (3 z³ + 4 z² - 1). In other words, this number is the least common integer divisor of all coefficients.

Monomial. Determine whether the same variable is in each of the terms of the polynomial. Assuming 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 a variable z. In addition, all coefficients are numbers that 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 common polynomial. Therefore, if factor-the binomial is not obvious, then you need to find at least one root. Select the free term of the polynomial; this is a coefficient without a variable. Now apply the method of substitution into the general expression of all integer divisors of the free term.

Consider: z^4 – 2 z³ + z² - 4 z + 4. Check if any of the integer factors of 4 are z^4 – 2 z³ + z² - 4 z + 4 = 0. By simple substitution, find z1 = 1 and z2 = 2, which means for brackets we can remove the binomials (z - 1) and (z - 2). To find the remaining expression, use sequential long division.

Students always ask: “Why can’t I use a calculator in the 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 inverse to the action of squaring.

√81= 9 9 2 =81

If you take the square root of a positive number and square the result, you 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 orally. Usually at school they teach a table of squares of natural numbers up to twenty. Knowing this table, it is easy to extract square roots from the numbers 121,144, 169, 196, 225, 256, 289, 324, 361, 400. From numbers greater than 400 you can extract them using the selection method using some tips. Let's try to look at this method with an example.

Example: Extract the root of the number 676.

We notice that 20 2 = 400, and 30 2 = 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.
This means that 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 = 6400, and 90 2 = 8100, then 80< √6889 < 90.
The number 9 is given by 3 2 and 7 2, then √6889 is equal to either 83 or 87.

Let's check: 83 2 = 6889.

Answer: √6889 = 83 .

If you find it difficult to solve using the selection method, you can factor the radical expression.

For example, find √893025.

Let's factor the number 893025, remember, you did this in the sixth grade.

We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.

More example: √20736. Let's factor the number 20736:

We get √20736 = √2 8 ∙3 4 = 2 4 ∙3 2 = 144.

Of course, factorization requires knowledge of divisibility signs and factorization skills.

And finally, there is rule for extracting square roots. Let's get acquainted with this rule with examples.

Calculate √279841.

To extract the root of a multi-digit integer, we divide it from right to left into faces containing 2 digits (the leftmost edge may contain one digit). We write it like this: 27’98’41

To obtain the first digit of the root (5), we take the square root of the largest perfect square contained in the first face on the left (27).
Then the square of the first digit of the root (25) is subtracted from the first face and the next face (98) is added to the difference (subtracted).
To the left of the resulting number 298, write the double digit of the root (10), divide by it the number of all tens of the previously obtained number (29/2 ≈ 2), test the quotient (102 ∙2 = 204 should be no 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 edge (41) is added to the difference (94).
To the left of the resulting number 9441, write the double product of the digits of the root (52 ∙2 = 104), divide the number of all tens of the number 9441 (944/104 ≈ 9) by this product, test the quotient (1049 ∙9 = 9441) should be 9441 and write it down (9) after the second digit of the root.

We received the answer √279841 = 529.

Extract similarly roots of decimal fractions. 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 a decimal fraction has an odd number of decimal places, the square root cannot be taken from it.

So now you have seen three ways to extract the root. Choose the one that suits you best and practice. To learn to solve problems, you need to solve them. And if you have any questions, .

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