The index laws only work in particular situations, and skip a condition and you'll get a tidy-looking answer that's quietly wrong. Here are the four that bite.
Trap 1 — the Bases Must Match
You can only add (or subtract) indices when the bases are the same. 2³ × 5⁴ is not 10⁷ or 2⁷ — the 2s and the 5s are different things, so there's nothing to combine. It just stays as 2³ × 5⁴. The laws are about counting repeats of one base; mix bases and the trick doesn't apply.
Trap 2 — When Multiplying, ADD (don't Multiply the Indices)
This is the big one. 3⁴ × 3² is 3⁶ (because 4 + 2 = 6), not 3⁸. Multiplying the indices is the power-of-a-power move, and that's a different question. Always ask first: am I multiplying powers (add) or raising a power to a power (multiply)?
Trap 3 — A⁰ = 1, Never 0
It's deeply tempting to say "zero of something is nothing, so 7⁰ = 0." Nope — 7⁰ = 1. Remember rung 1: 7³ ÷ 7³ = 7⁰, and a thing over itself is 1. Picture it that way and you'll never write 0 again.
Trap 4 — Coefficients Are Handled Separately
When there's a number out the front — a coefficient — keep it apart from the index work. 3 × 2⁴ × 5 × 2² = (multiply the numbers) 3 × 5 = 15, and (add the indices) 2⁴⁺² = 2⁶, giving 15 × 2⁶. Don't drag the coefficient into the index, and don't add the coefficients. Numbers with numbers, powers with powers.