IGCSE Maths / Powers and roots
Index laws with whole-number powers
Edexcel International GCSE Maths (4MA1), chapter 3: Powers and roots, specification point 1.4c-f. Free lesson.
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Index laws let you simplify powers without writing out every multiplication. They work by counting how many copies of the same base there are, so the base must match wherever you use a law.
How the three index laws work
A power tells you how many copies of its base are being multiplied. The index laws follow from joining or cancelling those copies. When multiplying powers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 2³ × 2² has five copies of 2, so it equals 2⁵. When dividing powers with the same base, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Copies on the top and bottom cancel. For a power of a power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ. There are n groups of m copies of a. Every law needs the same base on both sides. You cannot use these rules to simplify 2³ × 3².
How to simplify a calculation using more than one law
Apply each law to the part of the expression where it fits. For multiplication followed by division, combine the powers in the numerator first, then subtract the denominator's index.
Why a power of zero equals one
Dividing a non-zero number by itself gives 1. The division law also shows that the same division gives a power with index zero, so the two results must be equal. For any non-zero base a, dividing a power by itself gives aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰. But aᵐ ÷ aᵐ = 1, because anything divided by itself is 1. Therefore a⁰ = 1 for any non-zero base. For example, 7⁰ = 1.
How to avoid mixing up the index laws
Multiplying powers is not the same as raising a power to another power. Check whether the expression has two terms being multiplied or a bracket raised to a power. For 2³ × 2², the three copies of 2 join the two more copies, so the indices add: 2³ × 2² = 2⁵, not 2⁶. For (2³)², there are two groups of three copies, so the indices multiply: (2³)² = 2⁶. You can only add indices in a product when the bases match. For example, 2³ × 3² = 8 × 9 = 72, and the index law does not apply.
Use the laws in quick checks
For a division, subtract the indices. For a power raised to a power, multiply them. You can then work out the value if needed. 3⁸ ÷ 3⁵ = 3³ = 27. (4²)³ = 4⁶.