Question

Find the greatest common divisor (GCD) of $\sqrt{1,024}-\sqrt{225}$ and $\sqrt{18}\times \sqrt{8}$ using long division method.

Answer 1

First, find the square roots using prime factorization.

$1024=2^{10}$
$\Rightarrow \sqrt{1024}=\sqrt{2^{10}}=2^5=32$

$225=5\times 5\times 3\times 3$
$\Rightarrow \sqrt{225}=\sqrt{(5\times 3{)}^2}=5\times 3=15$

$\therefore \sqrt{1024}-\sqrt{225}=32-15=17$

$\sqrt{18}\times \sqrt{8}$

$=\sqrt{18\times 8}$

$=\sqrt{(2\times 3\times 3)\times (2\times 2\times 2)}$

$=\sqrt{2^2\times 2^2\times 3^2}$

$=2\times 2\times 3$

$=12$

$\therefore$ The greatest common divisor (GCD) of $\sqrt{1024}-\sqrt{225}$ and $\sqrt{50}\cdot \sqrt{18}$ is equivalent to GCD of $17$ and $12$.

By using long division method, we get:

GCD $=1$