Exponent Rules
Exponents are shorthand for repeated multiplication — and the rules are just bookkeeping. Add exponents when you multiply, subtract when you divide, multiply when you raise a power to a power. Zero and negative exponents fall out of the same patterns.
1 Why the rules work
An exponent counts factors: x3 = x · x · x. Every rule below is just counting factors — never memorize what you can re-derive.
Multiply → add the counts. Divide → subtract. Power of a power → multiply.
Why? x3 / x3 = x0, but it is also 1. And x2 / x5 = x−3 = 1/x3 — the extra factors fall to the denominator.
2 Worked examples
One per rule.
x3 · x4 = x7 — add the exponents: 3 + 4 = 7.
x6 / x2 = x4 — subtract: 6 − 2 = 4.
(x2)5 = x10 — multiply: 2 · 5 = 10.
2−3 = 1/23 = 1/8 — flip to the denominator.
3 Common mistakes
Three traps, each with the wrong version and the fix.
Wrong: x3 · x4 = x12. Right: multiply → add: x7. (Multiply only for a power of a power.)
Wrong: x6 / x2 = x8. Right: divide → subtract: x4.
Wrong: 2−3 = −8. Right: the minus moves the power across the fraction bar: 1/8.
4 Quick checks
Try these yourself, then reveal the answer.
5 Key points
Remember
- Multiply powers → add exponents: xa·xb = xa+b.
- Divide powers → subtract exponents: xa/xb = xa−b.
- Power of a power → multiply: (xa)b = xab.
- (cxa)b = cbxab — the exponent hits the coefficient too.
- x0 = 1 and x−a = 1/xa.