Mathematics — GCSE

Negative & Fractional Indices

Master the laws of indices for negative and fractional exponents to simplify complex expressions with confidence.

👁️ Visual 🖐️ Kinesthetic 👂 Auditory

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Objectives

Learning Objectives

Recall and apply the core laws of indices (multiply, divide, power of a power).
Understand and evaluate expressions with negative indices.
Understand and evaluate expressions with fractional indices, including unit and non-unit fractions.
Combine negative and fractional index rules to simplify and evaluate complex expressions.
Apply index laws to solve GCSE-style exam problems.

Theory — Recap

Recap: The Laws of Indices

Before tackling negative and fractional indices, let's recall the fundamental index laws that underpin everything else.

aᵐ × aⁿ = aᵐ⁺ⁿ
Multiplying powers → add the indices

aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Dividing powers → subtract the indices

(aᵐ)ⁿ = aᵐⁿ
Power of a power → multiply the indices

Key reminder: a⁰ = 1 for any non-zero value of a. This follows from the division rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰ = 1.

Theory — Negative Indices

Negative Indices

A negative index means we take the reciprocal. This follows from the division rule: a¹ ÷ a² = a⁻¹ = 1/a.

a⁻ⁿ = 1 / aⁿ
Flip the base to the denominator and make the index positive

In other words, raising something to a negative power puts it "underneath" a fraction line.

2⁻¹

= 1/2 = 0.5

5⁻²

= 1/5² = 1/25

3⁻³

= 1/3³ = 1/27

10⁻¹

= 1/10 = 0.1

Theory — Fractional Indices

Fractional Indices

A fractional index means taking a root. The denominator tells us which root, and the numerator tells us the power.

a^1/n = ⁿ√a
The denominator is the root

a^m/n = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
Root first, then power (or vice versa)

9^1/2

= √9 = 3

27^1/3

= ³√27 = 3

8^2/3

= (³√8)² = 2² = 4

16^3/4

= (⁴√16)³ = 2³ = 8

Theory — Combining Rules

Negative Fractional Indices

When an index is both negative and fractional, apply both rules: the negative means reciprocal, and the fraction means root and power.

a^−m/n = 1 / a^m/n = 1 / (ⁿ√a)ᵐ
Reciprocal + root + power

Strategy: Deal with the negative sign first (flip to a reciprocal), then handle the fractional index as normal.

4^−1/2

= 1/√4 = 1/2

27^−2/3

= 1/(³√27)² = 1/9

Theory — Worked Example

Worked Example

Simplify: 16^−3/4

Step 1: The negative index means reciprocal → 1 / 16^3/4

Step 2: The denominator 4 means fourth root → ⁴√16 = 2

Step 3: The numerator 3 means cube → 2³ = 8

Step 4: Put it together → 1/8

16^−3/4 = 1/8

Question 1 of 13

Evaluate 4⁻²

−16

1/16

−8

1/8

Question 2 of 13

Evaluate 10⁻³. Write your answer as a decimal.

Question 3 of 13

Evaluate 64^1/2

32

8

6

4

Question 4 of 13

Evaluate 125^1/3

Question 5 of 13

Evaluate 8^2/3

2

4

6

16

Question 6 of 13

Evaluate 32^2/5

Question 7 of 13

Evaluate 25^−1/2

−5

1/5

5

−1/5

Question 8 of 13

Evaluate 8^−2/3. Give your answer as a fraction.

Question 9 of 13

Match each expression to its value. Click a card, then click the box where it belongs.

1/9

4

1/2

8

3⁻²

Drop here

16^1/2

Drop here

4⁻^1/2

Drop here

32^3/5

Drop here

Question 10 of 13

Simplify: (x⁴)⁻²

x⁻⁸

x⁻²

x²

x⁻⁶

Question 11 of 13

Evaluate 16^−3/4. Give your answer as a fraction.

Question 12 of 13

Which of these is equivalent to 1 / ³√x ?

x⁻³

x^−1/3

x^1/3

x³

Question 13 of 13

Evaluate (49/9)^−1/2. Give your answer as a fraction.

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