The basic laws
An index (or exponent) tells you how many times a base is multiplied by itself: a3 = a × a × a. The laws of indices let you combine powers of the same base without expanding them.
Key formula
am × an = am+n
am ÷ an = am−n
(am)n = amn
a0 = 1, a−n = 1 / an, a1/n = n√a
Zero, negative and fractional indices
These three cases follow directly from the division law:
- Zero index: a0 = 1 for any a ≠ 0, because am ÷ am = a0 = 1.
- Negative index: a−n = 1 / an, so 4−2 = 1/16.
- Fractional index: am/n = n√(am), so 272/3 = (3√27)2 = 32 = 9.
Worked example
Simplify 23 × 25 ÷ 22.
Add the indices for multiplication, subtract for division: 3 + 5 − 2 = 6. So the answer is 26 = 64.
Now evaluate (32)3 = 36 = 729, and 161/2 = √16 = 4.
Solving index equations
When both sides can be written with the same base, equate the indices. To solve 2x = 32, write 32 = 25, so x = 5.
Remember
- The laws only combine powers of the same base.
- a0 = 1 (with a ≠ 0), not 0.
- A negative index means reciprocal, not a negative number.