How to find the root of a decimal. Root extraction: methods, examples, solutions

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 - numbers from which you can extract an integer Square root. 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.
You can write it down in the following way: √400 = √(25 x 16).

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
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
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. Verification with a calculator gives us the answer 5.92 - we were right.
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
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 be (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.

The calculation (or extraction) of the square root can be done in several ways, but all of them are not very simple. It's easier, of course, to resort to the help of a calculator. But if this is not possible (or you want to understand the essence of the square root), I can advise you to go the following way, its algorithm is as follows:

If you don’t have the strength, desire or patience for such lengthy calculations, you can resort to rough selection, its plus is that it is incredibly fast and, with due ingenuity, accurate. Example:

When I was in school (in the early 60s), we were taught to take the square root of any number. The technique is simple, outwardly similar to division by a column, but to state it here, it will take half an hour of time and 4-5 thousand characters of text. But why do you need it? Do you have a phone or other gadget, there is a calculator in nm. There is a calculator in every computer. Personally, I prefer to do this kind of calculation in Excel.

Often in school it is required to find square roots different numbers. But if we are used to using a calculator all the time for this, then there will be no such opportunity in exams, so you need to learn how to look for the root without the help of a calculator. And it is in principle possible to do so.

The algorithm is:

Look first at the last digit of your number:

For example,

Now you need to determine approximately the value for the root from the leftmost group

In the case when the number has more than two groups, then you need to find the root like this:

But the next number should be exactly the largest, you need to pick it up like this:

Now we need to form a new number A by adding to the remainder that was obtained above, the next group.

In our examples:

A column of najna, and when more than fifteen characters are needed, then computers and phones with calculators most often rest. It remains to check whether the description of the methodology will take 4-5 thousand characters.

Berm any number, from a comma we count pairs of digits to the right and left

For example, 1234567890.098765432100

A pair of digits is like a two-digit number. The root of a two-digit is one-to-one. We select a single-valued one, the square of which is less than the first pair of digits. In our case it is 3.

As when dividing by a column, under the first pair we write out this square and subtract from the first pair. The result is underlined. 12 - 9 = 3. Add a second pair of digits to this difference (it will be 334). To the left of the number of berms, the doubled value of the part of the result that has already been found is supplemented with a digit (we have 2 * 6 = 6), such that when multiplied by the not received number, it does not exceed the number with the second pair of digits. We get that the figure found is five. Again we find the difference (9), demolish the next pair of digits, getting 956, again write out the doubled part of the result (70), again add the necessary digit and so on until it stops. Or to the required accuracy of calculations.

Firstly, in order to calculate the square root, you need to know the multiplication table well. Most simple examples is 25 (5 by 5 = 25) and so on. If we take numbers more complicated, then we can use this table, where there are units horizontally and tens vertically.

There is good way how to find the root of a number without the help of calculators. To do this, you will need a ruler and a compass. The bottom line is that you find on the ruler the value that you have under the root. For example, put a mark near 9. Your task is to divide this number into an equal number of segments, that is, into two lines of 4.5 cm each, and into an even segment. It is easy to guess that in the end you will get 3 segments of 3 centimeters.

The method is not easy and will not work for large numbers, but it is considered without a calculator.

without the help of a calculator, the method of extracting the square root was taught in Soviet times at school in 8th grade.

To do this, you need to break a multi-digit number from right to left into faces of 2 digits :

The first digit of the root is the whole root of the left side, in this case, 5.

Subtract 5 squared from 31, 31-25=6 and add the next face to the six, we have 678.

The next digit x is selected to double the five so that

10x*x was the maximum, but less than 678.

x=6 because 106*6=636,

now we calculate 678 - 636 = 42 and add the next face 92, we have 4292.

Again we are looking for the maximum x, such that 112x*x lt; 4292.

Answer: the root is 563

So you can continue as long as you want.

In some cases, you can try to expand the root number into two or more square factors.

It is also useful to remember the table (or at least some part of it) - the squares of natural numbers from 10 to 99.

I propose a variant of extracting the square root into a column that I invented. It differs from the well-known, except for the selection of numbers. But as I found out later, this method already existed many years before my birth. The great Isaac Newton described it in his book General Arithmetic or a book on arithmetic synthesis and analysis. So here I present my vision and rationale for the algorithm of the Newton method. You don't need to memorize the algorithm. You can simply use the diagram in the figure as a visual aid if necessary.

With the help of tables, you can not calculate, but find, the square roots only from the numbers that are in the tables. The easiest way to calculate the roots is not only square, but also other degrees, by the method of successive approximations. For example, we calculate the square root of 10739, replace the last three digits with zeros and extract the root of 10000, we get 100 with a disadvantage, so we take the number 102 and square it, we get 10404, which is also less than the specified one, we take 103*103=10609 again with a disadvantage, we take 103.5 * 103.5 \u003d 10712.25, we take even more 103.6 * 103.6 \u003d 10732, we take 103.7 * 103.7 \u003d 10753.69, which is already in excess. You can take the square root of 10739 to be approximately equal to 103.6. More precisely 10739=103.629... . . Similarly, we calculate the cube root, first from 10000 we get approximately 25 * 25 * 25 = 15625, which is in excess, we take 22 * 22 * 22 = 10.648, we take a little more than 22.06 * 22.06 * 22.06 = 10735, which is very close to the given one.

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 numbers are used instead of letter designations variables.

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.

Taking the square root of a large number

We will now only talk about the case when the result of extracting the square root is an integer.

Case 1

So, let us by all means (for example, when calculating the discriminant) need to calculate the square root of 86436.

We will decompose the number 86436 into prime factors. We divide by 2, we get 43218; again we divide by 2, - we get 21609. The number is not divisible by 2 more. But since the sum of the digits is divisible by 3, then the number itself is divisible by 3 (generally speaking, it can be seen that it is also divisible by 9). . Once again we divide by 3, we get 2401. 2401 is not completely divisible by 3. Not divisible by five (does not end with 0 or 5).

We suspect divisibility by 7. Indeed, a ,

So, full order!

Case 2

Let us need to calculate . It is inconvenient to act in the same way as described above. Trying to factorize...

The number 1849 is not completely divisible by 2 (it is not even) ...

It is not completely divisible by 3 (the sum of the digits is not a multiple of 3) ...

It is not completely divisible by 5 (the last digit is not 5 or 0) ...

It is not completely divisible by 7, it is not divisible by 11, it is not divisible by 13 ... Well, how long will it take us to go through all the prime numbers like this?

Let's argue a little differently.

We understand that

We narrowed down the search. Now we sort through the numbers from 41 to 49. Moreover, it is clear that since the last digit of the number is 9, then it is worth stopping at options 43 or 47 - only these numbers, when squared, will give the last digit 9.

Well, here already, of course, we stop at 43. Indeed,

P.S. How the hell do we multiply 0.7 by 0.5?

You should multiply 5 by 7, ignoring the zeros and signs, and then separate, going from right to left, two decimal places. We get 0.35.

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Steps

Prime factorization

Calculating the square root manually

Using column division

Understanding the process

Taking the square root of a large number

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