A negative index means the reciprocal: a⁻ⁿ = 1/aⁿ, so 6⁻² = 1/36. A fractional index means a root: a^(1/n) is the nth root of a, so 9^(1/2) = 3. When the numerator is not 1, take the root first and then the power: a^(m/n) = (ⁿ√a)ᵐ. The usual index laws still apply.
Negative and fractional indices
- a⁻ⁿ = 1/aⁿ
- a^(1/n) = ⁿ√a
- a^(m/n) = (ⁿ√a)ᵐ
- (a/b)⁻ⁿ = (b/a)ⁿ
- aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ
Negative indices
A negative index means “one over”: take the reciprocal. 6⁻² = 1/6² = 1/36 and 5⁻¹ = 1/5. This follows from the division law: 2² ÷ 2⁵ = 2⁻³, and also equals 4/32 = 1/8. A negative index does not make the answer negative.
Fractional indices
A fractional index means a root, with the denominator telling you which root: a^(1/2) = √a and a^(1/3) = ∛a. When the numerator is not 1, it is a power as well. Take the root first, because the numbers stay smaller: 16^(3/4) = (⁴√16)³ = 2³ = 8.
Negative fractional indices
Deal with the three parts one at a time: the minus sign means reciprocal, the denominator means root, and the numerator means power. For 125^(−2/3): the cube root of 125 is 5, squared is 25, and the reciprocal is 1/25.
Fractions raised to a power
Apply the index to the numerator and the denominator separately: (8/27)^(2/3) = 4/9, because the cube roots are 2 and 3 and squaring gives 4 and 9. With a negative index, flip the fraction first: (2/3)⁻² = (3/2)² = 9/4.
Solving equations with indices
Write both sides as powers of the same base, then compare the indices. 3ˣ = 1/27 becomes 3ˣ = 3⁻³, so x = −3. For 4ˣ = 8, write both sides as powers of 2: 2²ˣ = 2³, so 2x = 3 and x = 3/2. Spotting the common base is the key step, so learn the small powers of 2, 3 and 5.
Simplifying with letters
The index laws work with negative and fractional indices exactly as with whole numbers. x^(3/2) × x^(5/2) = x⁴, because 3/2 + 5/2 = 4. To write an expression such as 1/√x as a power, use x^(−1/2).
Worked examples
Example 1. Work out 6⁻².
- Negative index: take the reciprocal, 1/6².
- 6² = 36.
Answer: 1/36.
Example 2. Work out 16^(3/4).
- Root first: the fourth root of 16 is 2.
- Then the power: 2³.
Answer: 8.
Example 3. Work out 125^(−2/3).
- Cube root of 125 = 5.
- Square: 25.
- Reciprocal: 1/25.
Answer: 1/25.
Example 4. Work out (8/27)^(2/3).
- Cube roots: ∛8 = 2 and ∛27 = 3, giving 2/3.
- Square: (2/3)² = 4/9.
Answer: 4/9.
Example 5. Simplify x^(3/2) × x^(5/2).
- Same base, so add the indices: 3/2 + 5/2 = 8/2.
Answer: x⁴.
Common mistakes
- Making the answer negative. 6⁻² = 1/36, not −36.
- Treating the fraction as division. 16^(1/2) is √16 = 4, not 16 ÷ 2 = 8.
- Taking the power before the root. 16^(3/4) is easier as (⁴√16)³ = 8 than as ⁴√4096.
- Forgetting to flip a fraction for a negative index. (2/3)⁻² = (3/2)² = 9/4.
- Writing a root as a fractional index the wrong way up. The cube root of x is x^(1/3), not x³ or x^(3/1).
- Multiplying indices when multiplying powers. x^(3/2) × x^(5/2) = x⁴: add the indices.
How negative and fractional indices are examined
Negative and fractional indices are Higher tier content in Edexcel GCSE (1MA1) and the International GCSE (4MA1), usually on non-calculator papers. You will be asked to:
- evaluate a number with a negative or fractional index, giving an exact fraction
- evaluate a combined negative fractional index
- simplify algebraic expressions with fractional powers
- solve an equation such as 2ˣ = 1/8
The indices worksheet includes Higher questions with answers. For extra practice, use the GCSE number revision engine.
This extends the laws of indices. At A-Level, the same ideas appear in indices and surds.
Frequently asked questions
What does a negative index mean?
The reciprocal. a⁻ⁿ = 1/aⁿ, so 2⁻³ = 1/8.
What does a fractional index mean?
A root. a^(1/2) is the square root of a, and a^(1/3) is the cube root.
How do you work out 27^(2/3)?
Take the cube root first, then square it.
Why does a negative index give a fraction?
Because dividing powers subtracts the indices. 2² ÷ 2⁵ = 2⁻³, and the same division gives 4/32 = 1/8, so 2⁻³ must equal 1/8.