Understanding Rational Exponents and Radicals Practice Test

Because $b^{m/n}$ means take the $n$th root first and then multiply by $m$, so $b^{m/n}=m\sqrt[n]{b}$.

Because $(b^{m/n})^n=b^m$, so $b^{m/n}$ must equal $\sqrt[n]{b^m}$; also $b^{m/n}=(b^{1/n})^m=(\sqrt[n]{b})^m$ by $(b^a)^c=b^{ac}$.

Because $\sqrt[n]{b^m}=(\sqrt[m]{b})^n$ since the $m$ and $n$ can be swapped in a radical, so all forms are equal.

Because $b^{m/n}=b^m/b^n$ by the quotient rule, so $b^{m/n}=b^{m-n}$ and that equals $\sqrt[n]{b^m}$.