Indices
Indices are also called powers. They are used to write repeated multiplication in a shorter form.
2³ means 2 × 2 × 2
2³ = 8
Index laws
GCSE Maths includes several important index laws. Learning these rules carefully makes algebra much easier.
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
Multiplying powers
Example 1: Multiply powers
Simplify x³ × x⁵.
Add the indices:
x³ × x⁵ = x⁸
Example 2: Another multiplication example
Simplify a² × a⁷.
a² × a⁷ = a⁹
Dividing powers
When dividing powers with the same base, subtract the indices.
Example 3: Divide powers
Simplify y⁹ ÷ y⁴.
Subtract the indices:
y⁹ ÷ y⁴ = y⁵
Example 4: Another division example
Simplify p⁶ ÷ p².
p⁶ ÷ p² = p⁴
Powers raised to powers
When a power is raised to another power, multiply the indices.
Example 5: Power raised to a power
Simplify (x³)².
Multiply the indices:
(x³)² = x⁶
Negative indices
A negative index means reciprocal.
a⁻¹ = 1/a
a⁻² = 1/a²
Example 6: Simplify a negative index
Simplify x⁻³.
x⁻³ = 1/x³
Fractional indices
Fractional indices are linked to roots.
a¹ᐟ² = √a
a¹ᐟ³ = ∛a
Example 7: Fractional index
Simplify 16¹ᐟ².
This means the square root of 16.
16¹ᐟ² = 4
A common mistake is multiplying indices when multiplying powers. For example, x² × x³ = x⁵, not x⁶.
Check whether the question is multiplying, dividing or raising a power to another power. Different index rules apply in each case.
Video explanation
A short Worthing Maths Tutor video explanation for indices can be embedded here later to improve student engagement and time on page.
Practice questions
- Simplify x⁴ × x².
- Simplify y⁹ ÷ y³.
- Simplify (a²)⁵.
- Write p⁻⁴ as a fraction.
- Simplify 81¹ᐟ².
Answers
- x⁶
- y⁶
- a¹⁰
- 1/p⁴
- 9
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