4MA1/2H Jan 2023, Question 26: Exponential Equations
Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2H (Higher) · Calculator · 5 marks
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The question, in short
Solve for n in the equation with powers of 10, 2 and 5 divided by 20² equal to 1, showing clear algebraic working.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Solution · 5 marks
- 110⁴ⁿ = (5 × 2)⁴ⁿ = 5⁴ⁿ × 2⁴ⁿ
- 220² = (2² × 5)² = 2⁴ × 5²
- 3The numerator is 5⁴ⁿ × 2⁴ⁿ × 2^(3n² − 15n) × 5^(2 − 4n)
- 4Powers of 5 in the numerator: 4n + 2 − 4n = 2, giving 5²
- 5Powers of 2 in the numerator: 4n + 3n² − 15n = 3n² − 11n
- 6So the left-hand side is (2^(3n² − 11n) × 5²) ÷ (2⁴ × 5²) = 2^(3n² − 11n − 4)
- 72^(3n² − 11n − 4) = 1 = 2⁰
- 83n² − 11n − 4 = 0
- 9(3n + 1)(n − 4) = 0
- 10n = −1/3 or n = 4
Answern = −1/3 or n = 4
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