Is there any condition while applying law of exponents?

Question

${[(-3)^2]}^\frac{1}{2}$ = ${(-3)^2}^\frac{1}{2}$ = $-3^1$ = $-3$

But counted other way it is

$9^\frac{1}{2} = \surd{9} = 3$

where I went wrong?

Answer 1

The second answer is the right one, i.e., ${[(-3)^2]}^\frac{1}{2}=3$. In your first attempt, you are trying to use the identity $(x^m)^n=x^{mn}$, which holds only true if both $x^m, x^n\in\mathbb{R}$. This is not the case here, as $\sqrt{-3}$ is not a real number.

As a general rule (i.e., for arbitrary $m,n\in\mathbb{R}$), the exponent laws are only valid for non-negative $x$.