# Number Properties Puzzle – Challenge 15

A mathematical challenge to tease your brain. Have a look at the solution to the puzzle if you can't solve it! Let's challenge your brain!

## 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!

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