Image: proof re fractional exponents

Question

Can someone help me prove that $(a^m)^{1/n} = (a^{1/n})^m$ per the textbook excerpt captured in the image?

Answer 1

Something like this?

$$(a^m)^{1/n} = a^{m \cdot (1/n)} = a^{(1/n) \cdot m} = (a^{1/n})^m$$

Answer 2

Write down the definition:

$$\sqrt[n]{a^m}:=a^{m/n}\stackrel{\text{exponents prop.}}=\left(a^{1/n}\right)^m=\left(\sqrt[n]a\right)^m$$