How to get out from under the root. Root extraction: methods, examples, solutions

21.10.2019

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

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

One of the simplest ways to calculate roots is to using a special table. What is it and how to use it correctly?

Using the table, you can find the square of any number from 10 to 99. At the same time, the rows of the table contain tens values, and the columns contain unit values. The cell at the intersection of a row and a column contains the square of a two-digit number. In order to calculate the square of 63, you need to find a row with a value of 6 and a column with a value of 3. At the intersection, we find a cell with the number 3969.

Since extracting the root is the inverse operation of squaring, to perform this action, you must do the opposite: first find the cell with the number whose radical you want to calculate, then determine the answer from the column and row values. As an example, consider the calculation of the square root of 169.

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

Similarly, you can calculate the roots of the cubic and n-th degree, using the appropriate tables.

The advantage of the method is its simplicity and the absence of additional calculations. The disadvantages are obvious: the method can only be used for a limited range of numbers (the number for which the root is found must be between 100 and 9801). In addition, it will not work if the given number is not in the table.

Prime factorization

If the table of squares is not at hand or with its help it was impossible to find the root, you can try decompose the number under the root into prime factors. Prime factors are those that can be completely (without remainder) divided only by itself or by one. Examples would be 2, 3, 5, 7, 11, 13, etc.

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

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

Heron's method

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

√R = √a + (R - a) / 2√a,

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

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

√111 = √121 + (111 - 121) / 2 ∙ √121 = 11 - 10 / 22 ≈ 10,55.

Now let's check the accuracy of the method:

10.55² = 111.3025.

The error of the method was approximately 0.3. If the accuracy of the method needs to be improved, you can repeat the steps described earlier:

√111 = √111,3025 + (111 - 111,3025) / 2 ∙ √111,3025 = 10,55 - 0,3025 / 21,1 ≈ 10,536.

Let's check the accuracy of the calculation:

10.536² = 111.0073.

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

Calculation of the root by division into a column

This method of finding the square root value is a little more complicated than the previous ones. However, it is the most accurate among other calculation methods without a calculator..

Let's say that you need to find the square root with an accuracy of 4 decimal places. Let's analyze the calculation algorithm using the example of an arbitrary number 1308.1912.

Divide the sheet of paper into 2 parts with a vertical line, and then draw another line from it to the right, slightly below the top edge. We write the number on the left side, dividing it into groups of 2 digits, moving to the right and left side from a comma. The very first digit on the left can be without a pair. If the sign is missing on the right side of the number, then 0 should be added. In our case, we get 13 08.19 12.
Let's pick the most big number, whose square will be less than or equal to the first group of digits. In our case, this is 3. Let's write it on the top right; 3 is the first digit of the result. At the bottom right, we indicate 3 × 3 = 9; this will be needed for subsequent calculations. Subtract 9 from 13 in a column, we get the remainder 4.
Let's add the next pair of numbers to the remainder 4; we get 408.
Multiply the number on the top right by 2 and write it on the bottom right, adding _ x _ = to it. We get 6_ x _ =.
Instead of dashes, you need to substitute the same number, less than or equal to 408. We get 66 × 6 \u003d 396. Let's write 6 on the top right, since this is the second digit of the result. Subtract 396 from 408, we get 12.
Let's repeat steps 3-6. Since the numbers carried down are in the fractional part of the number, it is necessary to put a decimal point on the top right after 6. Let's write the doubled result with dashes: 72_ x _ =. A suitable number would be 1: 721 × 1 = 721. Let's write it down as an answer. Let's subtract 1219 - 721 = 498.
Let's perform the sequence of actions given in the previous paragraph three more times to get the required number of decimal places. If there are not enough signs for further calculations, two zeros must be added to the current number on the left.

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

Bitwise calculation of the square root value

The method is highly accurate. In addition, it is quite understandable and it does not require memorizing formulas or a complex algorithm of actions, since the essence of the method is to select the correct result.

Let's extract the root from the number 781. Let's consider in detail the sequence of actions.

Find out which digit of the square root value will be the highest. To do this, let's square 0, 10, 100, 1000, etc. and find out between which of them the root number is located. We get that 10²< 781 < 100², т. е. старшим разрядом будут десятки.
Let's take the value of tens. To do this, we will take turns raising to the power of 10, 20, ..., 90, until we get a number greater than 781. For our case, we get 10² = 100, 20² = 400, 30² = 900. The value of the result n will be within 20< n <30.
Similarly to the previous step, the value of the units digit is selected. We alternately square 21.22, ..., 29: 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625, 26² = 676, 27² = 729, 28² = 784. We get that 27< n < 28.
Each subsequent digit (tenths, hundredths, etc.) is calculated in the same way as shown above. Calculations are carried out until the required accuracy is achieved.

Do you have dependency on the calculator? Or do you think that, except with a calculator or using a table of squares, it is very difficult to calculate, for example,.

It happens that schoolchildren are tied to a calculator and even multiply 0.7 by 0.5 by pressing the cherished buttons. They say, well, I still know how to calculate, but now I’ll save time ... There will be an exam ... then I’ll tense up ...

So the fact is that there will be plenty of “tense moments” at the exam anyway ... As they say, water wears away a stone. So on the exam, little things, if there are a lot of them, can knock you down ...

Let's minimize the number of possible troubles.

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.

Mathematics was born when a person became aware of himself and began to position himself as an autonomous unit of the world. The desire to measure, compare, calculate what surrounds you is what underlay one of the fundamental sciences of our days. At first, these were pieces of elementary mathematics, which made it possible to associate numbers with their physical expressions, later the conclusions began to be presented only theoretically (due to their abstractness), but after a while, as one scientist put it, "mathematics reached the ceiling of complexity when all 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.

How it all started

The first mention of the root, which is currently denoted as √, was recorded in the writings of the Babylonian mathematicians, who laid the foundation for modern arithmetic. Of course, they looked a little like the current form - the scientists of those years first used bulky tablets. But in the second millennium BC. e. they came up with an approximate calculation formula that showed how to take the square root. The photo below shows a stone on which Babylonian scientists carved the output process √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 the 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 subject 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 is not extracted without a remainder gives an irrational result.

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

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 is taken from an arbitrary number a, they wrote R 2 a. The “tick” √, familiar to the modern look, appeared only in the 17th century thanks to Rene Descartes.

Our days

Mathematically, the square root of y is the number z whose square is y. In other words, z 2 =y is equivalent to √y=z. However, this definition is relevant only for the 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 is valid for determining an algebraic root, the value of an 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 love for mathematics has only increased with the development of science, there are various manifestations of affection for it, not expressed in dry calculations. For example, along with such interesting events as the day of Pi, the holidays of the square root are also celebrated. They are celebrated nine times in a hundred years, and are determined according to the following principle: the numbers that denote the day and month in order must be the square root of the year. So, next time this holiday will be celebrated on April 4, 2016.

Properties of the square root on the field R

Almost all mathematical expressions have a geometric basis, this fate did not pass and √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 of the output is less than the subtracted one or even equals zero. The number of moves will eventually 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 an elementary function z=√y on the field of real numbers R, where y is greater than or equal to zero. Her chart looks like this:

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

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

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

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

3. The function takes the 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 by an ordinary power function.

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

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

The square root in the 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 degree root from 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 and with a negative discriminant got a solution. In C, for the square root, the same properties are relevant as in R, the only thing is that the restrictions on the root expression are removed.

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 availability 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 built into search engines Google and Nigma calculators. For example, if you need to find the largest integer factor that can be taken out of the sign of the cubic root for the number 250, then go to the Google website and enter the query "6 ^ 3" to check if it is possible to take out from under the sign root six. The search engine will show a result equal to 216. Alas, 250 cannot be divided without a remainder by this number. Then enter the query 5^3. The result will be 125, and this allows you to split 250 into factors of 125 and 2, which means taking it out of the sign root number 5 leaving there number 2.

Sources:

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

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

Instruction

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

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