Laws of Indices (Same Base, Same Indices)

An index shows how many times a base number is multiplied by itself. In 2³ the base is 2 and the index is 3, so 2³ means 2 × 2 × 2 = 8. Indices are also called powers or, in American books, exponents. The laws of indices are shortcuts for multiplying and dividing numbers written this way.

Rules for simplifying powers without writing them out. With the same base: multiply by adding the indices, 2³ × 2⁴ = 2⁷, and divide by subtracting them, 4⁵ ÷ 4³ = 4². With the same index: multiply or divide the bases and keep the index, 2³ × 5³ = 10³ and 6⁵ ÷ 3⁵ = 2⁵. These four are the GCSE core; the next lessons add negative, fractional and power-of-a-power rules.

It depends how you count them, which is why websites list anything from five to nine. This lesson covers four: add indices to multiply with the same base, subtract to divide, and multiply or divide the bases when the index is the same. Negative indices, a power of a power, fractional indices and the zero index come in the next lessons.

Add the indices and keep the base: 2³ × 2⁴ = 2⁷, because three 2s multiplied by four 2s is seven 2s. It works with negative indices too: 3⁻⁵ × 3⁷ = 3², since −5 + 7 = 2. And with three terms: 2⁴ × 2² × 2³ = 2⁹. Never multiply the indices here; that is the power-of-a-power rule.

Subtract the indices and keep the base: 4⁵ ÷ 4³ = 4² = 16, because five 4s divided by three 4s leaves two. The answer can have a negative index: 5⁴ ÷ 5⁶ = 5⁻², and 10⁵ ÷ 10² = 10³. Subtract the bottom index from the top one, in that order.

Combine the bases and keep the index. To multiply, multiply the bases: 2³ × 5³ = (2 × 5)³ = 10³, and 0.1⁸ × 10⁸ = 1⁸ = 1. To divide, divide the bases: 6⁵ ÷ 3⁵ = (6 ÷ 3)⁵ = 2⁵. The index never changes in either case.