Question

Evaluate $(\sqrt{484}-\sqrt{441}{)}^2$ using prime factorization.

Answer 1

Let's evaluate the prime factorization of $484$ and $441.$

$484$ is divisible by $4$ as last two digits is divisible by $4$.
$\frac{484}{4}=121=11\times 11$
$\therefore 484=4\times 11\times 11$
$\Rightarrow 484=2\times 2\times 11\times 11$
$\Rightarrow 484=(2\times 11)\times (2\times 11)$
​​​​​​​$\Rightarrow 484=(2\times 11{)}^2$
​​​​​​​$\Rightarrow \sqrt{484}=22$

Again, $441$ is divisible by $9.$

$\therefore 441=9\times 49$
$\Rightarrow 441=3\times 3\times 7\times 7$
$\Rightarrow 441=(3\times 7)\times (3\times 7)$
$\Rightarrow 441=(3\times 7{)}^2$
​​​​​​​$\Rightarrow \sqrt{441}=21$

$\therefore (\sqrt{484}-\sqrt{441}{)}^2=(22-21{)}^2=1^2=1$