Question

For a in $\mathbb{R}$, show that $\left | a \right |=\sqrt{a^2}$ . (Note that for $b \geq 0$ , $\sqrt{b}$ denotes the nonnegative square root of b.)

Answer 1

Since $\sqrt{b}$ is a non-negative no. Where b$\geq$0

So, if $a\geq0$ then

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

and if $a\leq0$ then

$\sqrt{a^{2}} = -a$   $\because \ a<0$

So, we can say that

$\sqrt{a^{2}} = a \quad if \ a\geq0 \qquad and \quad -a \quad if \ a< 0$

And we know that

$|a|= a \quad if \ a\geq0 \qquad and \quad -a \quad if \ a< 0$

Therefore

  $|a|= \sqrt{a^{2}}$