# Explain why the square root of a number is defined to be equal to that number to the 1/2 power?

Mar 13, 2018

see below

#### Explanation:

There are different ways of proving this, but I like this one...

The $\sqrt{x}$ is defined to be the opposite of ${x}^{2}$ meaning that
$\sqrt{{x}^{2}} = x$

Lets say we don't no the power of the square root jet

$\sqrt{x} = {x}^{n}$

We do know that ${\left({x}^{a}\right)}^{b} = {x}^{a \cdot b}$

Therefore, we can say that

$\sqrt{{x}^{2}} = x$

${\left({x}^{2}\right)}^{n} = x$

${x}^{2 \cdot n} = {x}^{1}$

Now, we can say that $2 n = 1$

which leads us to $n = \frac{1}{2}$