Which of the following numbers is both a perfect square and a perfect cube?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 64
Write each number as a product of primes. It is a perfect square when every power is even, and a perfect cube when every power is a multiple of 3; to be both, every power must be a multiple of 6.
\(64=2^{6}\): the power 6 is a multiple of 6, so 64 is both, \(8^{2}\) and \(4^{3}\)
\(72=2^{3}\times3^{2}\): the power 3 is odd, so it is not a square, and the power 2 is not a multiple of 3, so it is not a cube
\(128=2^{7}\): 7 is neither even nor a multiple of 3, so it is neither
\(200=2^{3}\times5^{2}\): the odd power 3 rules out a square and the power 2 rules out a cube