Index Laws

Grade 4

What index notation means

When we write 5³, the 5 is the base and the 3 is the index (also called a power or exponent). It just means 5 × 5 × 5. Index laws are rules that let you simplify expressions involving the same base without expanding everything out.

These rules only work when the bases are the same. You cannot directly combine 2³ × 3⁴ using index laws — the bases differ.

The three laws

Multiplying — add the powers. When you multiply two powers of the same base, add the indices.

aᵐ × aⁿ = aᵐ⁺ⁿ

Dividing — subtract the powers. When you divide two powers of the same base, subtract the indices.

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Power of a power — multiply the powers. When you raise a power to another power, multiply the indices.

(aᵐ)ⁿ = aᵐˣⁿ

There is also a fourth rule worth knowing: any base raised to the power zero equals 1. So 7⁰ = 1, and even 1000⁰ = 1.

The zero-power rule follows naturally from the division law: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and anything divided by itself is 1. So a⁰ = 1.

Worked examples

EXAMPLE
Simplify: (a) x⁵ × x³ (b) y⁸ ÷ y² (c) (z³)⁴
(a) Same base, multiplying — add the powers: x⁵ × x³ = x⁸. (b) Same base, dividing — subtract the powers: y⁸ ÷ y² = y⁶. (c) Power of a power — multiply: (z³)⁴ = z^(3×4) = z¹².
EXAMPLE
Write as a single power of 2: (2⁴ × 2³) ÷ 2⁵
Deal with the bracket first — multiply: 2⁴ × 2³ = 2⁷. Then divide: 2⁷ ÷ 2⁵ = 2⁷⁻⁵ = 2².

Don't get these mixed up

Multiplying law vs. power of a power is a common slip. x² × x³ = x⁵ (add), but (x²)³ = x⁶ (multiply). The bracket makes all the difference.
Nova
Try this one

Work out the value of \( 4^3 - 2^4 \). Show your working.

[3]

Nova's hint:

This involves evaluating two different powers and then subtracting.

Ask yourself: 'what does each power mean in terms of repeated multiplication?' Work out each term separately before subtracting. For example, \( 3^2 = 9 \) and \( 2^3 = 8 \).

Try to Solve it with Nova?