Arithmetic square root and its properties

2024 Author: Angel Austin | [email protected]. Last modified: 2023-12-17 05:14

We all studied arithmetic square roots in algebra class at school. It happens that if knowledge is not refreshed, then it is 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 schoolchildren, because we work with roots in grades 9, 10, and 11.

History of root and degree

Even in antiquity, and specifically in ancient Egypt, people needed degrees to perform operations on numbers. When there was no such concept, the Egyptians wrote down the product of the same number twenty times. But soon a solution to the problem was invented - the number of times that the number must be multiplied by itself began to be written in the upper right corner above it, and this form of recording has survived to this day.

And the history of the square root began about 500 years ago. It was designated in different ways, and only in the seventeenth century Rene Descartes introduced such a sign, which we use to this day.

What is a square root

Let's start by explaining what a square root is. The square root of some number c is a non-negative number that, when squared, will be equal to c. In this case, c is greater than or equal to zero.

To bring a number under the root, we square it and put the root sign over it:

32=9, 3=√9

Also, we cannot get the value of the square root of a negative number, since any number in a square is positive, that is:

c² ≧ 0, if √c is a negative number, then c^{2 < 0 - contrary to the rule.}

To quickly calculate square roots, you need to know the table of squares of numbers.

Properties

Let's consider the algebraic properties of the square root.

1) To extract the square root of the product, you need to take the root of each factor. That is, it can be written as the product of the roots of 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, write it as a quotient of their roots.

3) The value obtained by taking the square root of a number is always equal to the modulus of this number, since the modulus can only be positive:

√с2=∣с∣, ∣с∣ > 0.

4) To raise a root to any power, we raise to itradical expression:

(√с)⁴=√с^{4, for example:}

(√2)⁶ =√2^6=√64=8

5) The square of the arithmetic root of c is equal to this number itself:

(√s)2=s.

Roots of irrational numbers

Let's say the root of sixteen is easy, but how to take the root of 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 on our own. We can only compare it with other numbers. For example, take the root of 5 and compare it with √4 and √9. It is clear that √4 < √5 < √9, then 2 < √5 < 3. This means that the value of the root of five is somewhere between two and three, but there are a lot of decimal fractions between them, and picking 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 the 8th grade you will never be required to extract irrational numbers from the arithmetic square root. You only need 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) √17² - 8²

Remember the formula for the difference of squares:

√(17-8) (17+8)=√9 ×25

We know the property of the square arithmetic root - to extract the root from the product, you need to extract it from each factor:

√9 × √25=3 × 5=15

2) √3 (2√3 + √12)=2 (√3)2 + √36

Apply another property of the root - the square of the arithmetic root of a number is equal to this number itself:

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 - you can't do that!

We can't take the root of every term. There is no such rule, but it is confused with taking the root of each factor. If we had this entry:

√12 × 3, then it would be fair to write √12 × 3=√12 × √3.

And so we can only write:

√12 + 3=√15