IGCSE Maths / Powers and roots
Index laws with negative and zero powers
Edexcel International GCSE Maths (4MA1), chapter 3: Powers and roots, specification point 2.1a. Free lesson.
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The zero and negative index rules both come from the law for dividing powers with the same base. Knowing when the bases match helps you simplify expressions without using an index law incorrectly.
How to divide powers with the same base
When you divide powers with the same base, subtract the bottom index from the top index. This works because matching factors cancel. The bases must match, so a quotient with different letters cannot be simplified using this law. The rule is aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, x⁷ ÷ x³ = x⁴ because three matching x factors cancel.
Why a zero power equals one
Divide a power by itself. The division law gives an index of zero, but the identical powers also cancel to leave 1. Both results describe the same expression, so x⁰ = 1. The base must not be zero. For any non-zero base a, a⁰ = 1. A zero index does not mean the answer is zero.
What a negative power means
A negative index moves the power into the denominator of a fraction. It does not make the answer negative. This follows because multiplying aⁿ by a⁻ⁿ gives a⁰, which equals 1. For a non-zero base a, a⁻ⁿ = 1/aⁿ. For example, x⁻³ = 1/x³.
When the bases do not match
Index laws only apply when the bases match. With different bases, work out each power separately, then multiply or divide the results. Do not combine the bases and indices as if they were the same. 2³ × 3² is not 6⁵. Instead, 2³ × 3² = 8 × 9 = 72.
Simplifying a quotient with a negative index
Subtract the indices first. If this gives a negative index, rewrite it as a fraction with a positive index in the denominator.