A negative index means "reciprocal", not "negative number" — $2^{-3} = \dfrac{1}{8}$, a positive value. A fractional index tells you which root to take (the denominator) and which power to raise it to (the numerator).

If both sides can be rewritten as powers of the same base, then $a^x = a^y \implies x=y$ — you can simply equate the exponents.

Worked Example

Solve $9^{x+1} = 27^{x}$.

1. Rewrite both sides using base $3$: $9 = 3^2$, $27 = 3^3$.
2. Substitute: $(3^2)^{x+1} = (3^3)^{x}$.
3. Apply the power-of-a-power law, multiplying the exponent across the whole bracket: $3^{2x+2} = 3^{3x}$.
4. Equate exponents (same base): $2x+2 = 3x$.
5. Solve: $x = 2$.

The power distributes over multiplication and division (but never over addition or subtraction):

Deal with the three parts of the index separately, in a sensible order: take the root given by the denominator, raise to the power given by the numerator, then apply the reciprocal from the negative sign.

Worked Example

Evaluate $8^{-\frac{2}{3}}$.

1. The negative sign means reciprocal: $8^{-\frac{2}{3}} = \dfrac{1}{8^{\frac{2}{3}}}$.
2. The denominator of the fractional index ($3$) means cube root: $\sqrt[3]{8}=2$.
3. The numerator ($2$) means square that result: $2^2=4$.
4. Combine: $8^{-\frac{2}{3}} = \dfrac{1}{4}$.

Apply the laws one at a time, in whatever order removes brackets and combines matching bases first — there's no single fixed order, but working from the innermost brackets outward usually keeps the numbers smallest and the working cleanest.

For example, $\dfrac{(2x^2)^3}{4x^4}$ should be tackled by expanding the power-of-a-power first — $(2x^2)^3 = 2^3 x^6 = 8x^6$ — before dividing, giving $\dfrac{8x^6}{4x^4} = 2x^2$. Trying to divide before expanding the bracket usually leads to a mismatched, harder expression.