We all at school studied arithmetic square roots in algebra lessons. It happens that if the knowledge is not refreshed, then they are quickly forgotten, the same with the roots. This article will be useful to eighth-graders who want to refresh their knowledge in this area, and other students, because we work with roots in 9th, 10th, and 11th grades.
History of Root and Degree
Even in antiquity, and specifically in Ancient Egypt, people needed degrees to complete operations on numbers. When there was no such thing, the Egyptians recorded a work of the same number twenty times. But soon a solution was invented - the number of times that the number must be multiplied by itself was written in the upper right corner above it, and this form of recording has been preserved to this day.
And the story of the square root began about 500 years ago. It was designated differently, and it was only in the seventeenth century that Rene Descartes introduced the sign that we still use today.
What is the square root
Let's start by explaining what the square root is. The square root of some number c is a non-negative number that, when squared, will be equal to c. Moreover, c is greater than or equal to zero.
To make a number under the root, we square it and put a root sign above it:
3 2 = 9, 3 = √9
Also, we cannot get the square root value of a negative number, since squared any number is positive, that is:
c 2 ≥ 0, if √ c is a negative number, then c 2 <0 - which contradicts the rule.
To quickly calculate the square roots, you need to know the table of squares of numbers.
The properties
Consider the algebraic properties of the square root.
1) To extract the square root of the product, you need to extract the root of each factor. That is, it can be painted as the product of the roots of the factors:
√ac = √a × √c, for example:
√36 = √4 × √9
2) When extracting a root from a fraction, it is necessary to extract the root separately from the numerator and denominator, that is, paint their roots as a quotient.
3) The value obtained by extracting the square root of the number is always equal to the module of this number, since the module can only be positive:
√ 2 = ∣∣, ∣∣> 0.
4) To raise the root to some extent, we raise the root expression into it:
(√) 4 = √ 4 , for example:
(√2) 6 = √2 6 = √64 = 8
5) The square of the arithmetic root of c is equal to this very number:
(√ s) 2 = s.
The roots of irrational numbers
Suppose it is easy to calculate the root of sixteen, but how to extract the root from numbers like 7, 10, 11?
A number whose root is an infinite non-periodic fraction is called irrational. We cannot extract the root from it ourselves. We can only compare it with other numbers. For example, take the root of number 5 and compare it with √4 and √9. It is clear that √4 <√5 <√9, then 2 <√5 <3. Therefore, the value of the root of five is somewhere between the two and the three, but there are many decimal fractions between them, and selecting each is a dubious way finding the root.
You can do this operation on a calculator - this is the easiest and fastest way, but in grade 8 you will never need to extract irrational numbers from the arithmetic square root. It is only necessary to remember the approximate values of the root of two and the root of three:
√2 ≈ 1.4,
√3 ≈ 1.7.
Examples
Now, based on the properties of the square root, we will solve several examples:
1) √17 2 - 8 2
We recall the formula of the difference of squares:
√ (17-8) (17 + 8) = √9 × 25
We know the property of a square arithmetic root - to extract a root from a product, you need to extract it from each factor:
√9 × √25 = 3 × 5 = 15
2) √3 (2√3 + √12) = 2 (√3) 2 + √36
We apply one more property of the root - the square of the arithmetic root of the number is equal to this very number:
2 × 3 + 6 = 12
Important! Often, when starting to work and solve examples with arithmetic square roots, students make the following mistake:
√12 + 3 = √12 + √3 - this cannot be done!
We cannot extract the root from each term. There is no such rule, but it is confused with extracting the root from each factor. If we had such a record:
√12 × 3, it would be fair to write √12 × 3 = √12 × √3.
And so we can only write:
√12 + 3 = √15