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# Square and square roots

We are interested in this post in defining two opposite categories of numbers, square and square roots. The square of a number is obtained from this number by itself. While the square root is obtained by solving a second-order algebraic equation. Let’s find out these two numbers together.

## The definition of square a square roots

For centuries humans have used square roots, at least geometrically. Indeed, let ABC be a triangle such that the measurement of side $AB$ is equal to the measurement of side $AC$, for example, 1 centimeter in length. What is the measure of side $BC$. Let $a$ be the length of $BC$. According to the Pythagorean relation, $a^2=1^2+1^2=2$. And after that mathematicians showed that a number $a$ satisfying $a^2=2$ is not a rational number. It is a real number. In this section, we will discuss this category of numbers.

### The square of numbers

The square of a number is obtained by multiplying this number with its ego. This means that the square of the number $a$ is $a\times a$. We denote \begin{align*} a\times a:=a^2.\end{align*} In fact, a square is just a number $a$ with exponent $2$.

For example, the square of $4$ is $4^2=16$. The square of $1$ is $1$. The square of any negative number is$$(-a)\times (-a)=(-a)^2=a^2.$$The square of $(-3)$ is $(-3)^2=3^2=9$.

The square of the product and fraction numbers are $$(ab)^2=a^2\times b^2,\qquad \left(\frac{c}{d}\right)^2=\frac{c^2}{d^2}.$$

### The square root of  positive numbers

A square root of $b$ is a number $x$ whose square is $b$, that is \begin{align*}x^2=b.\end{align*} The square root of $b$ is denoted by $\sqrt{b}$ or sometimes $b^{\frac{1}{2}}$.

Example: Find the square root of 25. It is a number $x$ such that $x^2=25$. Hence $x=5$. Thus\begin{align*} \sqrt{25}=5.\end{align*}We also have  $\sqrt{1}=1$ and $\sqrt{0}=0$. If $a$ and $b$ are positive numbers then$$\sqrt{ab}= \sqrt{a}\times \sqrt{b}.$$

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