How to solve square root. Square Root Conversion

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.

Of small numbers that are perfect squares 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.

You just have to remember that if decimal has an odd number of decimal places, the square root cannot be extracted from it exactly.

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, sign up for my lessons.

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Exponentiation involves multiplying a given number by itself a certain amount once. For example, raising the number 2 to the fifth power would look like this:

The number that needs to be multiplied by itself is called the base of the power, and the number of multiplications is called its exponent. Raising to a power corresponds to two opposite actions: finding the exponent and finding the base.

Root extraction

Finding the base of a power is called root extraction. This means that you need to find the number that needs to be raised to the power n to get the given one.

For example, it is necessary to extract the 4th root of the number 16, i.e. to determine, you need to multiply by itself 4 times to ultimately get 16. This number is 2.

This arithmetic operation recorded using special sign– radical: √, above which the exponent is indicated on the left.

Arithmetic root

If the exponent is an even number, then the root can be two numbers with the same absolute value, but c is positive and negative. So, in the example given, these could be the numbers 2 and -2.

The expression must be unambiguous, i.e. have one result. For this purpose the concept was introduced arithmetic root, which can only be a positive number. An arithmetic root cannot be less than zero.

Thus, in the example discussed above, only the number 2 will be the arithmetic root, and the second answer option - -2 - is excluded by definition.

Square root

For some degrees, which are used more often than others, there are special names that are originally associated with geometry. We are talking about raising to the second and third powers.

To the second power the length of a side of a square when you need to calculate its area. If you need to find the volume of a cube, the length of its edge is raised to the third power. Therefore it is called the square of the number, and the third is called the cube.

Accordingly, the root of the second degree is called square, and the root of the third degree is called cubic. The square root is the only root that is not written with an exponent above the radical:

So, the arithmetic square root of a given number is the positive number that must be raised to the second power to get the given number.

The area of ​​a square plot of land is 81 dm². Find his side. Suppose the side length of the square is X decimeters. Then the area of ​​the plot is X² square decimeters. Since, according to the condition, this area is equal to 81 dm², then X² = 81. The length of a side of a square is a positive number. A positive number whose square is 81 is the number 9. When solving the problem, it was necessary to find the number x whose square is 81, i.e. solve the equation X² = 81. This equation has two roots: x 1 = 9 and x 2 = - 9, since 9² = 81 and (- 9)² = 81. Both numbers 9 and - 9 are called the square roots of 81.

Note that one of square roots X= 9 is positive number. It is called the arithmetic square root of 81 and is denoted √81, so √81 = 9.

Arithmetic square root of a number A is a non-negative number whose square is equal to A.

For example, the numbers 6 and - 6 are square roots of the number 36. However, the number 6 is an arithmetic square root of 36, since 6 is a non-negative number and 6² = 36. The number - 6 is not an arithmetic root.

Arithmetic square root of a number A denoted as follows: √ A.

The sign is called the arithmetic sign square root; A- called a radical expression. Expression √ A read like this: arithmetic square root of a number A. For example, √36 = 6, √0 = 0, √0.49 = 0.7. In cases where it is clear that we're talking about about an arithmetic root, they briefly say: “the square root of A«.

The act of finding the square root of a number is called square rooting. This action is the reverse of squaring.

You can square any number, but you can't extract square roots from any number. For example, it is impossible to extract the square root of the number - 4. If such a root existed, then, denoting it with the letter X, we would get the incorrect equality x² = - 4, since there is a non-negative number on the left and a negative number on the right.

Expression √ A only makes sense when a ≥ 0. The definition of square root can be briefly written as: √ a ≥ 0, (√A)² = A. Equality (√ A)² = A valid for a ≥ 0. Thus, to ensure that the square root of a non-negative number A equals b, i.e. in the fact that √ A =b, you need to check that the following two conditions are met: b ≥ 0, b² = A.

Square root of a fraction

Let's calculate. Note that √25 = 5, √36 = 6, and let’s check whether the equality holds.

Because and , then the equality is true. So, .

Theorem: If A≥ 0 and b> 0, that is, the root of the fraction is equal to the root of the numerator divided by the root of the denominator. It is required to prove that: and .

Since √ A≥0 and √ b> 0, then .

On the property of raising a fraction to a power and the definition of a square root the theorem is proven. Let's look at a few examples.

Calculate using the proven theorem .

Second example: Prove that , If A ≤ 0, b < 0. .

Another example: Calculate .

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Square Root Conversion

Removing the multiplier from under the root sign. Let the expression be given. If A≥ 0 and b≥ 0, then using the product root theorem we can write:

This transformation is called removing the factor from the root sign. Let's look at an example;

Calculate at X= 2. Direct substitution X= 2 in the radical expression leads to complex calculations. These calculations can be simplified if you first remove the factors from under the root sign: . Now substituting x = 2, we get:.

So, when removing the factor from under the root sign, the radical expression is represented in the form of a product in which one or more factors are squares of non-negative numbers. Then apply the product root theorem and take the root of each factor. Let's consider an example: Simplify the expression A = √8 + √18 - 4√2 by taking out the factors in the first two terms from under the root sign, we get:. We emphasize that equality valid only for A≥ 0 and b≥ 0. if A < 0, то .

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 abstractness), 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 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 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, as 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.