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 radical number it is necessary to break it on the edges so that the comma is between the edges.
Example. Find the value √0.00956484.
You just have to remember that if decimal has odd number 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.
website, when copying material in full or in part, a link to the source is required.
Square square plot 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 square root sign; 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 equal to the root from 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 .
.
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 a factor from under the root sign, represent the radical expression in the form of a product in which one or more factors are squares 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 when A≥ 0 and b≥ 0. if A < 0, то .
Raising a number to a power is a shortened form of writing the operation of multiple multiplication, in which all factors are equal to the original number. And extracting the root means reverse operation- determination of the multiplier that must be involved in the multiple multiplication operation so that the result is a radical number. Both the exponent and the root exponent indicate the same thing - how many factors there should be in such a multiplication operation.
You will need
- Internet access.
Instructions
- If you need to apply both the operation of extracting the root and raising it to a power to a number or expression, reduce both operations into one - raising to a power with a fractional exponent. The numerator of the fraction must contain an exponent, and the denominator must contain a root. For example, if you need to square a cubic root, then these two operations will be equivalent to one raising a number to the ⅔ power.
- If the conditions require squaring root with an exponent equal to two, this is not a calculation task, but a test of your knowledge. Use the method from the first step and you will get the fraction 2/2, i.e. 1. This means that the result of squaring the square root of any number will be that number itself.
- Square if necessary root with an even exponent, there is always the possibility of simplifying the operation. Since two (numerator) fractional indicator degree) and any even number (denominator) is common divisor, then after simplifying the fraction, one will remain in the numerator, which means that it is not necessary to raise to a power in calculations, it is enough to extract root with half the exponent. For example, squaring the sixth root of eight can be reduced to extracting cube root, because 2/6=1/3.
- To calculate the result for any exponent of the root, use, for example, the calculator built into search engine Google. This is perhaps the most easy way calculations if you have access to the Internet from your computer. A generally accepted substitute for the sign of the operation of exponentiation is this “lid”: ^. Use it when entering a search query into Google. For example, if you want to square root fifth power from the number 750, formulate the query as follows: 750^(2/5). After entering it, the search engine, even without pressing the send button to the server, will show the calculation result accurate to seven decimal places: 750^(2 / 5) = 14.1261725.
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 it 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.
