Extracting the root is the reverse operation of raising a power. That is, taking 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, but the numbers may be large.
Extracting the root of a number
Taking the square root of a number is easy. Moreover, this can be done not immediately, but gradually. For example, take the expression √256. Initially, it is difficult for an ignorant person to give an answer right away. Then we will do it step by step. First, we divide by just the number 4, from which we take the selected square as the root.
Let's represent: √(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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Extracting a complex root
The square root cannot be calculated from negative numbers, because any squared number is a positive number!
A complex number is the number i, which squared is equal to -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 a negative number, but this already applies to higher mathematics, not school mathematics.
Let's consider an example of such a root extraction: √(-49)=7*√(-1)=7i.
Online root calculator
Using our calculator, you can calculate the extraction of a number from the square root:
Converting Expressions Containing a Root Operation
The essence of transforming 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 give an example, √625. Let's divide the radical expression by the number 5. We get √(125 5), repeat the operation √(25 25), but we know that 25 is 52. Which means the answer will be 5*5=25.
But there are numbers for which the root cannot be calculated using this method and you just need to know the answer or have a table of squares at hand.
√289=√(17*17)=17
Bottom line
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It's time to sort it out root extraction methods. They are based on the properties of roots, in particular, on the equality, which is true for any non-negative number b.
Below we will look at the main methods of extracting roots one by one.
Let's start with the simplest case - extracting roots from natural numbers using a table of squares, a table of cubes, etc.
If tables of squares, cubes, etc. If you don’t have it at hand, it’s logical to use the method of extracting the root, which involves decomposing the radical number into prime factors.
It is worth special mentioning what is possible for roots with odd exponents.
Finally, let's consider a method that allows us to sequentially find the digits of the root value.
Let's get started.
Using a table of squares, a table of cubes, etc.
In the simplest cases, tables of squares, cubes, etc. allow you to extract 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 specific row and a specific column, it allows you to compose 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 cell 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 ones there is a cell with the number 6,889, 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. accordingly from the numbers in these tables. Let us explain the principle of their use when extracting roots.
Let's say we need to extract the nth root of the number a, while the number a is contained in the table of nth powers. Using 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 to use a cube table to extract the cube root of 19,683. We find the number 19,683 in the table of cubes, from it we find that this number is the cube of the number 27, therefore, 
.
It is clear that tables of nth powers are very convenient for extracting roots. However, they are often not at hand, and compiling them requires some time. Moreover, it is often necessary to extract roots from numbers that are not contained in the corresponding tables. In these cases, you have to resort to other methods of root extraction.
Factoring a radical number into prime factors
A fairly convenient way to extract the root of a natural number (if, of course, the root is extracted) is to decompose the radical number into prime factors. His the point is this: after that it is quite easy to represent it as a power with the desired exponent, which allows you to obtain the value of the root. Let us clarify this point.
Let the nth root of a natural number a be taken and its value equal b. In this case, the equality a=b n is true. The number b, like any natural number, can be represented as the product of all its prime factors p 1 , p 2 , …, p m in the form p 1 ·p 2 ·…·p m , and the radical number a in this case is represented as (p 1 ·p 2 ·…·p m) n . Since the decomposition of a number into prime factors is unique, the decomposition of the radical number a into prime factors will have the form (p 1 ·p 2 ·…·p m) n, which makes it possible to calculate the value of the root as.
Note that if the decomposition into prime factors of a radical number a cannot be represented in the form (p 1 ·p 2 ·…·p m) n, then the nth root of such a number a is not completely extracted.
Let's figure this out when solving examples.
Example.
Take the square root of 144.
Solution.
If you look at the table of squares given in the previous paragraph, you can clearly see that 144 = 12 2, from which it is clear that the square root of 144 is equal to 12.
But in light of this point, we are interested in how the root is extracted by decomposing the radical number 144 into prime factors. Let's 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 the properties of the roots, the solution could be formulated a little differently: .
Answer:
To consolidate the material, consider the solutions to two more examples.
Example.
Calculate the value of the root.
Solution.
The prime factorization of the radical number 243 has the form 243=3 5 . Thus, 
.
Answer:
Example.
Is the root value an integer?
Solution.
To answer this question, let's factor the radical 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 expansion cannot be represented as a cube of an integer, since the power of the prime factor 7 is not a multiple of three. Therefore, the cube root of 285,768 cannot be extracted completely.
Answer:
No.
Extracting roots from fractional numbers
It's time to figure out how to extract the root of a fractional number. Let the fractional radical number be written as p/q. According to the property of the root of a quotient, the following equality is true. From this equality it follows rule for extracting the root of a fraction: The root of a fraction is equal to the quotient of the root of the numerator divided 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 the common fraction 25/169?
Solution.
Using the table of squares, we find that the square root of the numerator of the original fraction is equal to 5, and the square root of the denominator is equal to 13. Then 
. This completes the extraction of the root of the common fraction 25/169.
Answer:
The root of a decimal fraction or mixed number is extracted after replacing the radical numbers with ordinary fractions.
Example.
Take the cube root of the decimal fraction 474.552.
Solution.
Let's imagine the original decimal fraction as 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 
. All that remains is to complete the calculations 
.
Answer:
.
Taking the root of a negative number
It is worthwhile to dwell on extracting roots from negative numbers. When studying roots, we said that when the root exponent is an odd number, then there can be a negative number under the root sign. We gave these entries the following meaning: for a negative number −a and an odd exponent of the root 2 n−1, 
. This equality gives rule for extracting odd roots from negative numbers: to extract the root of a negative number, you need to take the root of the opposite positive number, and put a minus sign in front of the result.
Let's look at the example solution.
Example.
Find the value of the root.
Solution.
Let's transform the original expression so that there is a positive number under the root sign: 
. Now replace the mixed number with an ordinary fraction: 
. We apply the rule for extracting the root of an ordinary fraction: 
. It remains to calculate the roots in the numerator and denominator of the resulting fraction: 
.
Here is a short summary of the solution: 
.
Answer:
.
Bitwise determination of 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 in this case there is a need to know the meaning 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 sequentially obtain a sufficient number of digit values of the desired number.
The first step of this algorithm is to find out what the most significant bit of the root value is. To do this, the numbers 0, 10, 100, ... are sequentially raised to the power n until the moment when a number exceeds the radical number is obtained. Then the number that we raised to the power n at the previous stage will indicate the corresponding most significant digit.
For example, consider this step of the algorithm when extracting the square root of five. 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 the most significant digit will be the ones digit. The value of this bit, as well as the lower ones, will be found in the next steps of the root extraction algorithm.
All subsequent steps of the algorithm are aimed at sequentially clarifying the value of the root by finding the values of the next bits of the desired value of the root, starting with the highest one and moving to the lowest ones. For example, the value of the root at the first step turns out to be 2, at the second – 2.2, at the third – 2.23, and so on 2.236067977…. Let us describe how the values of the digits are found.
The digits are found by searching through 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 radical 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 these points using the same example of extracting the square root of five.
First we find the value of the units digit. We will go through the values 0, 1, 2, ..., 9, calculating 0 2, 1 2, ..., 9 2, respectively, until we get a value greater than the radical number 5. It is convenient to present all these calculations in the form of a table: 
So the value of the units digit is 2 (since 2 2<5
, а 2 3 >5 ). Let's move on to finding the value of the tenths place. In this case, we will square the numbers 2.0, 2.1, 2.2, ..., 2.9, comparing the resulting values with the radical number 5: 
Since 2.2 2<5
, а 2,3 2 >5, then the value of the tenths place is 2. You can proceed to finding the value of the hundredths place: 
This is how the next value of the root of five was found, it is equal to 2.23. And so you can continue to find values: 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 determine the most significant 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 determine its value. 
Since 10 3<2 151,186
, а 20 3 >2 151.186, then the value of the tens place is 1. Let's move on to units. 
Thus, the value of the ones digit is 2. Let's move on to tenths. 
Since even 12.9 3 is less than the radical number 2 151.186, then the value of the tenths 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 accurate 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, the ones we studied above are sufficient.
Bibliography.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).
There are several methods for calculating the square root without a calculator.
How to find the root of a number - 1 way
- One method is to factor the number under the root. These components, when multiplied, form a radical value. The accuracy of the result depends on the number under the root.
- For example, if you take the number 1,600 and start factoring it, the reasoning will be structured as follows: this number is a multiple of 100, which means it can be divided by 25; since the root of the number 25 is taken, the number is square and suitable for further calculations; when dividing, we get another number - 64. This number is also square, so the root can be extracted well; After these calculations, under the root, you can write the number 1600 as the product of 25 and 64.
- One of the rules for extracting a root states that the root of the product of factors is equal to the number that is obtained by multiplying the roots of each factor. This means that: √(25*64) = √25 * √64. If we take the roots from 25 and 64, we get the following expression: 5 * 8 = 40. That is, the square root of the number 1600 is 40.
- But it happens that the number under the root cannot be decomposed into two factors, from which the whole root is extracted. Typically this can only be done for one of the multipliers. Therefore, most often it is not possible to find an absolutely exact answer in such an equation.
- In this case, only an approximate value can be calculated. Therefore, you need to take the root of the multiplier, which is a square number. This value is then multiplied by the root of the second number that is not the square term of the equation.
- It looks like this, for example, let’s take the number 320. It can be decomposed into 64 and 5. You can extract the whole root from 64, but not from 5. Therefore, the expression will look like this: √320 = √(64*5) = √64*√5 = 8√5.
- If necessary, you can find the approximate value of this result by calculating
√5 ≈ 2.236, therefore √320 = 8 * 2.236 = 17.88 ≈ 18. - Also, the number under the root can be decomposed into several prime factors, and the same ones can be taken out from under it. Example: √75 = √(5*5*3) = 5√3 ≈ 8.66 ≈ 9.
How to find the root of a number - 2nd method
- Another way is to do long division. Division occurs in a similar way, but you just need to look for square numbers, from which you can then extract the root.
- In this case, we write the square number on top and subtract it on the left side, and the extracted root from below.
- Now the second value needs to be doubled and written from the bottom right in the form: number_x_=. The gaps must be filled in with a number that is less than or equal to the required value on the left - just like in normal division.
- If necessary, this result is again subtracted from the left. Such calculations continue until the result is achieved. You can also add zeros until you reach the desired number of decimal places.
Root formulas. Properties of square roots.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
In the previous lesson we figured out what a square root is. It's time to figure out which ones exist formulas for roots what are properties of roots, and what can be done with all this.
Formulas of roots, properties of roots and rules for working with roots- this is essentially the same thing. There are surprisingly few formulas for square roots. Which certainly makes me happy! Or rather, you can write a lot of different formulas, but for practical and confident work with roots, only three are enough. Everything else flows from these three. Although many people get confused in the three root formulas, yes...
Let's start with the simplest one. Here she is:
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
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