These eight questions are original and mix all four skills in the chapter. Attempt each one with full working before opening the answer, and compare your method as well as your result.
Questions
Question 1. Evaluate 16^(3/4).
Answer
Take the fourth root first: ⁴√16 = 2. Then 2³ = 8.
Question 2. Evaluate 27^(−2/3), giving the answer as a fraction.
Answer
The negative power means a reciprocal: 27^(−2/3) = 1 ÷ 27^(2/3).
27^(2/3) = (∛27)² = 3² = 9, so the answer is 1/9.
Question 3. Solve 4^x = 32^(x−1).
Answer
Write both sides as powers of 2: 2^(2x) = 2^(5(x−1)) = 2^(5x−5).
Equate the powers: 2x = 5x − 5, so 3x = 5 and x = 5/3.
Check: 4^(5/3) = 2^(10/3), and 32^(2/3) = 2^(10/3).
Question 4. Express (2 + √3)² in the form a + b√3, then write 1 ÷ (2 + √3) without a surd in the denominator.
Answer
(2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3. The middle term 2 × 2 × √3 must not be left out.
For the fraction, multiply by the conjugate 2 − √3: 1 × (2 − √3) ÷ [(2 + √3)(2 − √3)] = (2 − √3) ÷ (4 − 3) = 2 − √3.
Question 5. Rationalise 5 ÷ (√6 − 1).
Answer
The conjugate is √6 + 1. Denominator: (√6 − 1)(√6 + 1) = 6 − 1 = 5.
Numerator: 5(√6 + 1). The fraction is 5(√6 + 1) ÷ 5 = √6 + 1.
Question 6. Given log₂ 3 = p, express log₂ 36 in terms of p.
Answer
36 = 4 × 9, so log₂ 36 = log₂ 4 + log₂ 9 = 2 + 2 log₂ 3 = 2 + 2p.
Question 7. Solve log₅(2x + 1) = 2.
Answer
Convert to an index: 2x + 1 = 5² = 25, so 2x = 24 and x = 12.
Domain check: 2(12) + 1 = 25 > 0, so the answer is accepted.
Question 8. Solve log₂(x − 1) + log₂(x + 1) = 3.
Answer
Combine: log₂(x² − 1) = 3, so x² − 1 = 8 and x² = 9. Hence x = 3 or x = −3.
The domain needs x − 1 > 0, so x > 1. Therefore x = 3 and x = −3 is rejected.
Check: log₂ 2 + log₂ 4 = 1 + 2 = 3.
If you got these wrong
- Questions 1 to 3: read applying index laws with fractional powers.
- Questions 4 and 5: study simplifying and rationalising surds.
- Questions 6 to 8: work through using logarithm laws with domain checks and solving exponential and logarithmic equations.
Note each slip in the mistake log and retry the question a few days later. The timed practice session builder can set up a timed round.
To have a teacher watch your working, see online one-to-one Additional Mathematics tuition, or go back to the chapter overview.