Q.Simplify: $(2^{-1})^{-3}$.

Apply the power-of-a-power law first (multiply the indices), then interpret the resulting sign.

$(2^{-1})^{-3}=2^{(-1)\times(-3)}=2^3=8$.

By the power-of-a-power law, multiply the indices:

since a negative times a negative is positive. Evaluating: $2^3=8$.

Check (independent recomputation): first evaluate the inner power directly, $2^{-1}=\dfrac12$; then $\left(\dfrac12\right)^{-3}=\left(\dfrac21\right)^3=2^3=8$ — matches.

Mishandling the sign when multiplying two negative exponents (wrongly treating $(-1)\times(-3)$ as $-3$ instead of $+3$), which would give the wrong reciprocal answer $\frac18$ instead of $8$.