Limits of Exponent Laws I have recently learned (discovered) that the exponent law $b^{mn} = {(b^m)}^n$ is not universally applicable. To demonstrate, if it were we could conclude that $(-1)^{\frac{3}{2}}$ (or by extension -1 to any power) is equal to 1.
$(-1)^{\frac{3}{2}} = (-1)^{2*\frac{3}{4}}$ 
= $((-1)^2)^{\frac{3}{4}}$ 
= $1^{\frac{3}{4}}$ 
= $1$
Under exactly what circumstances (natural/integer/rational/real bases or exponents) are the various exponent laws applicable? For example, can I use the above law ( $b^{mn} = {(b^m)}^n$ ) with a negative base when the exponents are required to be integers?

Basically, the one thing you need to remember is : $b^m$ does not have a signification when the two following condition are simultaneously met :
-b is negative
-m is not an integer
So in your example, $(-1)^{\frac{3}{2}}$ does not have any signification. Hence obviously, you should not write it.
Just remember that and you can only write correct equalities.
