Power of power rule of exponents

The power of a power rule says $(x^a)^b = x^{a \times b}$.
It makes sense because $(x^a)^b$ means multiplying $x^a$ by itself $b$ times, which stacks up $a \times b$ copies of $x$.

So you just multiply the two exponents.
For example, $(3^2)^4 = 3^{2 \times 4} = 3^8$.
There is no need to expand everything out.

The rule also covers fractional exponents, and a fraction stands for a root.
Since $\frac{1}{2}$ means a square root, $(x^6)^{\frac{1}{2}} = x^{6 \times \frac{1}{2}} = x^3$.
A cube root would be a power of $\frac{1}{3}$, and so on.

A useful trick is to rewrite the base as a power of a smaller prime.
Since $8 = 2^3$, you can write $8^n = (2^3)^n = 2^{3n}$.
Turning everything into the same base lets you compare or simplify exponents easily.

Simplify $(3^2)^4$.
Multiply the exponents: $(3^2)^4 = 3^{2 \times 4}$.
So it is $3^8$.

Write $(32^8)^{\frac{1}{2}}$ as a power of $2$.
First, $(32^8)^{\frac{1}{2}} = 32^{8 \times \frac{1}{2}} = 32^4$.
Then since $32 = 2^5$, $32^4 = (2^5)^4 = 2^{20}$.

Express $8^n$ as a power of $2$.
Rewrite the base: $8 = 2^3$, so $8^n = (2^3)^n$.
By the power-of-a-power rule, that is $2^{3n}$.