Number Properties Puzzle – Challenge 15

Challenge:

Which of the following is neither a perfect square number nor a perfect cube number? (A perfect square is a number that can be expressed as the product of two equal integers. For example, 4 is a perfect square number. 2×2=4
A perfect cube is the result of multiplying a number three times by itself. For example, 27 is a perfect cube number. 3×3×3=27)

A- \(2^9\)

B- \(3^8\)

C- \(4^7\)

D- \(5^6\)

E- \(6^5\)

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Let’s analyze each option:
\(2^9 = (2^3)^3\), therefore, \(2^9\) is a perfect cube number.
\(3^8 = (3^4)^2\), therefore, \(3^8\) is a perfect square number.
\(4^7 = (2^2)^7 = 2^{14} = (2^7)^2\), therefore, \(4^7\) is a perfect square number.
\(5^6 = (5^3)^2\), therefore, \(5^6\) is a perfect square and perfect cube number.
\(6^5\) cannot be written with different base. So, this is the answer!