Question

Find the square root of the following number by the prime factorisation method.
$\frac{5625}{3969}$

Answer 1

The factors of 5625 is $3\times 3\times 5\times 5\times 5\times 5$ and the factors of 3969 is $3\times 3\times 3\times 3\times 7\times 7$. Therefore,

$\frac{5625}{3969}=\frac{3\times 3\times 5\times 5\times 5\times 5}{3\times 3\times 3\times 3\times 7\times 7}$

Taking square root both sides,

$\sqrt{\frac{5625}{3969}}=\sqrt{\frac{3\times 3\times 5\times 5\times 5\times 5}{3\times 3\times 3\times 3\times 7\times 7}}$

$=\sqrt{\frac{3^2\times 5^4}{3^4\times 7^2}}$

$=\sqrt{{\left(\frac{3\times 5^2}{3^2\times 7}\right)}^2}$

$=\frac{3\times 5^2}{3^2\times 7}$

$=\frac{75}{63}$

Therefore, the square root of $\frac{5625}{3969}$ is $\frac{75}{63}$.