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MathidiaMathematics Academy
Indices and standard form
Lesson 6 of 75
Introduction

Index laws

Objective. Multiply and divide powers with integer indices.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Index laws are rules that simplify expressions involving powers of the same base. When multiplying powers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. When dividing, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. When raising a power to another power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ. For example, 3⁴ × 3² = 3⁶, since 4 + 2 = 6, which can be checked as 81 × 9 = 729 = 3⁶. These laws only apply directly when the base is identical, so 2³ × 5² cannot be simplified using them.

Worked example

Simplify the expression x⁷ × x³ ÷ x⁴ using index laws.

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Illustrated card showing ?
  1. 1Multiply the first two terms: x⁷ × x³ = x⁷⁺³ = x¹⁰.
  2. 2Now divide by x⁴ using the subtraction rule.
  3. 3x¹⁰ ÷ x⁴ = x¹⁰⁻⁴.
  4. 4x¹⁰⁻⁴ = x⁶.

Answer: x⁶

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Common mistake. Pupils often multiply the indices instead of adding them when multiplying powers, writing 2³ × 2² as 2⁶ rather than 2⁵. Remember: multiplying powers means adding indices, not multiplying them.

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