General index laws
Objective. Apply index laws to numerical and algebraic expressions.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Index laws are the rules for combining powers of the same base, extended here to algebraic terms with letters and mixed coefficients. When multiplying powers of the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ; when dividing, subtract them: aᵐ ÷ aⁿ = aᵐ⁻ⁿ; and for a power of a power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ. For algebraic expressions like (3x²)³, apply the power to both the coefficient and the index: 27x⁶. Any base to the power zero equals 1, and negative indices give reciprocals.
Worked example
Simplify (4x⁵)² ÷ 2x⁴ using the index laws fully.
- 1Expand the bracket first: (4x⁵)² = 16x¹⁰.
- 2Divide the coefficients: 16 ÷ 2 = 8.
- 3Subtract the indices: x¹⁰ ÷ x⁴ = x⁶.
- 4Combine the results: 8x⁶.
Answer: 8x⁶
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Practice
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Start practisingCommon mistake. Pupils often simplify (2x³)² as 2x⁶ instead of applying the power to the coefficient too, giving 4x⁶. Always raise both the number and the letter term to the outer power.
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