IGCSE Maths / Powers and roots
Evaluating fractional powers
Edexcel International GCSE Maths (4MA1), chapter 3: Powers and roots, specification point 1.4c. Free lesson.
Read the revision notes · IGCSE Maths past papers
A fractional power tells you to take a root and then raise the result to a power. Reading the fraction correctly helps you evaluate expressions accurately and keep the numbers manageable.
What the fraction in a power means
In a fractional power, the denominator tells you which root to take and the numerator tells you the ordinary power. So you can rewrite the expression using a root, then work out the power. a^(m/n) = (ⁿ√a)^m For example, a denominator of 3 means a cube root. A numerator of 2 means square the result.
How to evaluate a fractional power
Take the root first, then apply the power. The other order gives the same answer, but taking the root first usually keeps the numbers smaller.
A power of 1/2 means a square root
A fractional power is an instruction about what to do to the number. It is not a fraction that you multiply the number by. In particular, a power of 1/2 means take the square root, not take half. 9^(1/2) = √9 = 3 Half of 9 is 9 × 1/2 = 4.5, which is a different calculation.
Avoid multiplying by the fraction
Do not treat the fraction in the power as a multiplier. For 125^(2/3), the denominator means cube root and the numerator means square. This gives a much different result from multiplying 125 by 2/3. 125^(2/3) = (³√125)² = 5² = 25 The cube root is 5 because 5³ = 125.
What to do with a negative fractional power
A negative power means take the reciprocal, so deal with the minus sign first. Then use the denominator as the root and the numerator as the power. A negative power flips the expression to 1 divided by the same power with a positive index.
Check the order on another example
For 8^(2/3), the denominator gives a cube root and the numerator gives a square. Taking the root first makes the calculation straightforward. ³√8 = 2, then 2² = 4.