PK 2015 Q6
6. Which of the following is neither a square nor a cube?
- A \(2^9\)
- B \(3^{10}\)
- C \(4^{11}\)
- D \(5^{12}\)
- E \(6^{13}\)
Pink Kangaroo 2015, Question 6. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E \(6^{13}\)
Using the index law \(a^{mn} = (a^m)^n\), we see \(2^9 = (2^3)^3\), a cube; \(3^{10} = (3^5)^2\), a square; \(4^{11} = (2^2)^{11} = (2^{11})^2\), a square; and \(5^{12} = (5^4)^3\), a cube. This leaves \(6^{13}\) which is neither a square nor a cube since 6 is neither a square nor a cube, and 13 is not divisible by 2 nor by 3.
Solution © UK Mathematics Trust, from the PK 2015 solutions.