Question

Determine the GCD of $\sqrt{729}-\sqrt{324}$ and $\sqrt{32}\times \sqrt{18}$ using long division method.

Answer 1

First, find the square roots using prime factorization.

$729=3^6$
$\Rightarrow \sqrt{729}=\sqrt{3^6}=3^3=27$

$324=2\times 2\times 3\times 3\times 3\times 3$
$\Rightarrow \sqrt{324}=\sqrt{(2\times 3\times 3{)}^2}=2\times 9=18$

$\therefore \sqrt{729}-\sqrt{324}=27-18=9$

$\sqrt{32}\times \sqrt{18}$
$=\sqrt{32\times 18}$
$=\sqrt{(4\times 8)\times (2\times 9)}$
$=\sqrt{2^2\times 4^2\times 3^2}$
$=2\times 4\times 3$
$=24$

$\therefore$ GCD of $\sqrt{729}-\sqrt{324}$ and $\sqrt{32}\cdot \sqrt{18}$ is equivalent to GCD of $9$ and $24.$

Using long division method,
GCD $=3$