A fractional index is a root and a power combined: a^(m/n) means take the n-th root of a, then raise the result to the power m. Once you read the fraction that way, the ordinary index laws still apply.
This lesson belongs to SPM Additional Mathematics indices, surds and logarithms. The next lesson, simplifying and rationalising surds, applies the same idea to roots.
What do the laws say?
Keep these six laws together and practise them as a set.
| Law | Example |
|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ |
| (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ |
| a^(1/n) = ⁿ√a | 49^(1/2) = 7 |
| a^(m/n) = (ⁿ√a)ᵐ | 8^(2/3) = 2² = 4 |
| a^(−n) = 1 ÷ aⁿ | 2^(−3) = 1/8 |
Worked example: evaluate a product of fractional powers
Evaluate 27^(2/3) × 4^(−1/2).
First factor. The denominator 3 is the cube root and the numerator 2 is the square. ∛27 = 3, then 3² = 9.
Second factor. The negative sign means a reciprocal, and the 1/2 means square root. 4^(−1/2) = 1 ÷ √4 = 1/2.
Multiply: 9 × 1/2 = 4.5, or 9/2.
The mistake that costs marks
The common slip is to read 8^(2/3) as 8 × 2 ÷ 3. It looks plausible because the fraction sits in the power, but a power is never a multiplier.
| Step | Wrong | Right |
|---|---|---|
| Read the fraction | 8 × 2/3 = 16/3 | Root 3, power 2 |
| Root | (skipped) | ∛8 = 2 |
| Power | (skipped) | 2² = 4 |
| Answer | 16/3 | 4 |
A quick test: 8^(1/3) must be the number that multiplies by itself three times to give 8. That is 2, not 8 ÷ 3.
Solving equations by matching bases
When the unknown is in the power, write both sides with the same base. Then equal bases force equal powers.
Solve 4^(x+1) = 8^x.
- Write 4 = 2² and 8 = 2³.
- Then (2²)^(x+1) = (2³)^x, so 2^(2x+2) = 2^(3x).
- The bases match, so 2x + 2 = 3x, which gives x = 2.
- Check: 4³ = 64 and 8² = 64.
Fractional powers also appear with algebra. Simplify (16x⁸)^(3/4): 16^(3/4) = (⁴√16)³ = 2³ = 8, and (x⁸)^(3/4) = x⁶. So the answer is 8x⁶.
If you want to test your own questions, the word problem structure worksheet helps you set out a multi-step problem before starting.
Check yourself
Simplify 32^(3/5) ÷ 2^(−2), then solve 9^x = 27^(x−1).
Answer
32^(3/5) = (⁵√32)³ = 2³ = 8. Also 2^(−2) = 1/4, so 8 ÷ (1/4) = 8 × 4 = 32.
For the equation, write 9 = 3² and 27 = 3³: 3^(2x) = 3^(3(x−1)) = 3^(3x−3).
So 2x = 3x − 3 and x = 3. Check: 9³ = 729 and 27² = 729.
What to study next
Roots of numbers that are not perfect powers need their own tools. Continue with simplifying and rationalising surds, then try the chapter practice set.
If you want a teacher to go through index questions with your own mistakes, see online one-to-one Additional Mathematics tuition. The mistake log helps you record which step slips.