# What 3-digit number is a perfect square and a perfect cube?

Jun 28, 2015

$729 = {9}^{3} = {27}^{2}$
The answer must be ${a}^{6}$ for some integer $a$ due to the properties of exponentiation. Only ${3}^{6} = 729$ lies in the required range.
${3}^{6} = {\left({3}^{2}\right)}^{3} = {9}^{3}$
${3}^{6} = {\left({3}^{3}\right)}^{2} = {27}^{2}$