Question

$729$ is a perfect square number, and is a square of .

Answer 1

Prime Factorization of $729$ can be denoted as:
$729=3\times 3\times 3\times 3\times 3\times 3$
$\Rightarrow 729=3^6$
$\Rightarrow 729=3^{3\times 2}$
$\Rightarrow 729=(3^3{)}^2$ $[\because a^{m\times n}=(a^m{)}^n]$
$\Rightarrow \underset{–}{729={27}^2}$ $[\because 3^3=3\times 3\times 3=27]$

$\Rightarrow \sqrt{729}=\sqrt{{27}^2}$ (Taking square root both the sides)
$\Rightarrow {729}^{\frac{1}{2}}=({27}^2{)}^{\frac{1}{2}}$
$\Rightarrow {729}^{\frac{1}{2}}={27}^{2\times \frac{1}{2}}(\because (a^m{)}^n=a^{m\times n})$
$\Rightarrow \underset{–}{\sqrt{729}=27}$

Hence, $729$ is a perfect square number, and $729$ is a square of $27$.