Unpacking exponents

An exponent is a shorthand for repeated multiplication: $x^3 = x \cdot x \cdot x$.
When you multiply powers of the same base, you add the exponents: $x^a \cdot x^b = x^{a+b}$.
When you divide them, you subtract: $\frac{x^a}{x^b} = x^{a-b}$.

Two more rules handle powers of powers and products.
A power of a power multiplies the exponents: $(x^a)^b = x^{ab}$.
And a power spreads over everything in a product: $(ab)^n = a^n b^n$, so $(2x)^3 = 2^3 x^3 = 8x^3$.

A negative exponent means take the reciprocal: $x^{-n} = \frac{1}{x^n}$.
This also works in reverse, so $\frac{1}{x^{-n}} = x^n$ — a negative exponent in the denominator moves up top and becomes positive.

Two special cases round things out.
Any nonzero base to the power $0$ is $1$, so $x^0 = 1$.
A fractional exponent is a root: $x^{a/b} = \sqrt[b]{x^a}$.
For example, $27^{4/3} = (3^3)^{4/3} = 3^4 = 81$.

Hard-looking expressions come apart one rule at a time.
Take $\frac{(2x)^3}{(3x)^{-2}}$.
First spread the powers: $(2x)^3 = 8x^3$, and the negative exponent flips up to give $(3x)^2 = 9x^2$.
Then multiply and add exponents: $8x^3 \cdot 9x^2 = 72x^5$.

Simplify $(4y)^2$.
A power spreads over a product: $(4y)^2 = 4^2 \cdot y^2$.
So $(4y)^2 = 16y^2$.

Simplify $27^{4/3}$.
Write $27$ as $3^3$, then use the power-of-a-power rule: $(3^3)^{4/3} = 3^{3 \times \frac{4}{3}} = 3^4$.
So $27^{4/3} = 81$.

Simplify $(5a^{-3})^{-2}$.
Spread the outer power: $(5a^{-3})^{-2} = 5^{-2} \cdot (a^{-3})^{-2}$.
The power of a power multiplies exponents, so $(a^{-3})^{-2} = a^{6}$, and $5^{-2} = \frac{1}{25}$.
So the result is $\frac{a^6}{25}$.