IGCSE Maths / Powers and roots
Evaluating negative powers revision notes
Edexcel International GCSE Maths (4MA1), chapter 3: Powers and roots, specification point 1.4c. Free lesson.
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A negative power means take the reciprocal, then use the matching positive power. This helps you evaluate powers correctly and avoid mistaking a negative power for a negative answer.
Why a⁰ = 1
The zero-power rule follows from the law for dividing powers with the same base: subtract the powers. For example, 5³ ÷ 5³ = 5³⁻³ = 5⁰. But a number divided by itself is 1, so 5⁰ must also equal 1. The power tells you what to do, not what the answer is. For a non-zero number a, a⁰ = 1. A power of zero gives 1, not 0.
What a negative power means
A negative power turns a number into its reciprocal, which means one divided by that number. The minus sign belongs to the power, so it does not make the answer negative. a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8. This is positive and smaller than 1. This works because 2³ × 2⁻³ = 2⁰ = 1, so 2⁻³ must turn 8 into 1.
Why 5⁻² is not −25
The minus sign in 5⁻² is attached to the power, not to the 5. It tells you to take the reciprocal of 5². A minus sign in front of a number, as in −25, is a different operation. 5⁻² = 1/5² = 1/25. The result is positive and smaller than 1, not negative.
How to evaluate a negative power of a fraction
For a fraction, the negative power flips the fraction over, then the power becomes positive. You then raise the top and bottom to that positive power. A fraction smaller than 1 becomes bigger than 1 when flipped.
Evaluate a negative power of a whole number
Use the same reciprocal rule for a whole number. Write it as one over the number with a positive power, then calculate. A negative power gives a positive result when the base is positive. The result is smaller than 1 for a positive whole-number base greater than 1.