Powers and Indices: Laws, Formulas and Examples

A power is shorthand for repeated multiplication, written aⁿ, where a is the base and n the index. For example 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Eight index laws simplify powers, and the topic is taught in Year 9.

The Eight Index Laws

Multiply with the same base: aᵐ × aⁿ = aᵐ⁺ⁿ (2³ × 2⁴ = 128). Divide: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (2⁵ ÷ 2² = 8). Power of a power: (aᵐ)ⁿ = aᵐⁿ (2³)² = 64. Power of a product and of a quotient, zero index a⁰ = 1, negative index a⁻ⁿ = 1 ÷ aⁿ (2⁻³ = 1/8), and fractional index 8^(1/3) = 2.

Why a⁰ = 1

Any number divided by itself is 1, and 2³ ÷ 2³ = 2⁰, so 2⁰ = 1.

Square Roots

A square root undoes squaring: √36 = 6 because 6 × 6 = 36. Using prime factors, √576 = 2³ × 3 = 24. Note that √(9 + 16) = 5 but √9 + √16 = 7.

Worked Examples

• (3² × 3⁴) ÷ 3³ = 27
• (2⁻²)⁻³ = 64
• 5³ × 5⁻¹ = 25
• √144 + 5² = 37
• −2⁴ = −16 but (−2)⁴ = 16

Summary

A negative index gives a fraction, not a negative number. Algonova has helped more than 1,000,000 students in 97 countries, with a free maths diagnostic of about 60 minutes.