Indices (negative & fractional) and roots

When multiplying like bases the indices ADD; students often multiply them (or vice versa), e.g. mishandling $p^a \times p^b$.

Failing to express numbers like $9$, $27$ or $81$ as powers of $3$, so the equation can't be solved by matching exponents.

Thinking $a^{-2}$ is negative rather than $\tfrac{1}{a^2}$; or forgetting the root in a fractional index.

For $a^{m/n}$ you take the $n$th root and raise to the $m$ — you do not multiply. So $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$, not $16 \times \tfrac{3}{4} = 12$. Multiplying the base by the index is a classic slip that gives a wildly wrong size of answer.

Any non-zero base to the power $0$ equals $1$: $a^0 = 1$, so $7^0 = 1$ and $(3x)^0 = 1$. Students often write $0$ instead. It commonly surfaces after a division that cancels fully, e.g. $a^3 \div a^3 = a^0 = 1$.