Square and square root of numbers



In this article, we are interested in the square and the square root 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*}

Examples: the sequare of $4$ is $4^2=16$. The square of $1$ is $1$.

The square of any negative number is 

$$ (-a)\times (-a)=(-a)^2=a^2.$$

Example: the square of $(-3)$ is $(-3)^2=3^2=9$.

The square of 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 numbes

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 postive numbers then 

$$ \sqrt{ab}= \sqrt{a}\times \sqrt{b}.$$

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