64 is a square number (8^2 ) and a cube number (4^3 ). Are there … - Vidyadip

Question

64 is a square number $\left(8^2\right)$ and a cube number $\left(4^3\right)$. Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

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Answer

Yes, there are other numbers that are both squares and cubes. For example :
$\begin{array}{l}729=9^3 \text { (cube number) }=27^2 \text { (perfect square) } \\4096=16^3 \text { (cube number) }=64^2 \text { (perfect square) }\end{array}$
General Rule: The sixth power of any number (i.e., $n^6$ ) is both a square and a cube. i.e.
$\begin{array}{l}1^6=1 \\2^6=64 \\3^6=729 \\4^6=4096 \\5^6=15,625\end{array}$

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