A shorter way to write repeated multiplication

Suppose a number doubles, then doubles again, then doubles again. Starting with 1, that gives 2 × 2 × 2 = 8. A power keeps track of the repeated factor without writing it out every time.

We read 2^³ as “2 to the power of 3”. The small raised number is the index, also called the exponent. It counts the factors, not the multiplication signs.

The position matters: 5^⁴ means four factors of 5, not the two-digit number 54.

Working out a power

For 2^⁵, multiply five factors of 2, one at a time: 2 × 2 = 4, then 4 × 2 = 8, then 8 × 2 = 16, and finally 16 × 2 = 32.

Increasing the index by 1 adds another factor, not another 1 to the answer. For powers of 2, each step doubles the value. Changing the base changes every factor.

Multiplying the base by the index is a different calculation. For example, 3^⁴ = 3 × 3 × 3 × 3 = 81, whereas 3 × 4 = 12.

Squares and cubes

Two powers have familiar names. 4^² is “4 squared”: a square with side length 4 cm has area 4 × 4 = 16 cm². 4^³ is “4 cubed”: a cube with edge length 4 cm has volume 4 × 4 × 4 = 64 cm³.

An index of 1 means just one factor: 7^¹ = 7. For any positive whole-number index, powers of 1 stay at 1, and powers of 0 stay at 0.

Writing a product as powers

Going the other way, count the equal factors. The product 7 × 7 × 7 is 7^³. If there are different factors, collect each kind separately.

Reordering the factors doesn't change the product. There are two factors of 2 and three of 5, so each base gets its own index.