IMC 2017 Q14
14. For what value of \(x\) is \(64^x\) equal to \(512^5\)?
- A 6
- B 7.5
- C 8
- D 16
- E 40
Intermediate Mathematical Challenge 2017, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 7.5
Note that \(64 = 2^6\) and \(512 = 2^9\). Therefore \((2^6)^x = (2^9)^5\). So \(2^{6x} = 2^{45}\).
Hence \(6x = 45\), that is \(x = 7.5\).
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).