Powers with the same base

An index counts the multiplies

2³ is not 2 × 3. The small number counts how many copies of the base are MULTIPLIED together: 2³ = 2 × 2 × 2 = 8. Get that meaning solid and the index laws stop being rules to memorise — they're just counting. Multiply a³ by a²? That's three a's times two a's — five a's: a⁵, so you ADD indices. Divide a⁶ by a²? Two a's cancel — a⁴: you SUBTRACT. Raise a power to a power, (a²)³? Three copies of two a's — six: you MULTIPLY. And a⁰ = 1 because a³ ÷ a³ has to be 1, and the law says it's a⁰. Whenever a law feels doubtful, write out the multiplies — the law is sitting right there.

• 2³ = 2×2×2 = 8. (2×3 = 6 is a different animal.)
• a³ × a² = (a·a·a)(a·a) = a⁵ — add.
• a⁶ ÷ a² = a⁴ — subtract. And a³ ÷ a³ = a⁰ = 1.
• (a²)³ = a²·a²·a² = a⁶ — multiply.