Question

Find the square roots of the following numbers by Prime-factorization method.

(i) $441$

(ii) $784$

(iii) $4096$

(iv) $7056$

Answer 1

(a) $441$

Prime Factorisation:

$441=3\times 3\times 7\times 7$

So, $\sqrt{441}=\sqrt{3\times 3\times 7\times 7}$

$=3\times 7$

$=21$

Hence, $\sqrt{441}=21$

(b) $784$

Prime Factorisation:

$784=2\times 2\times 2\times 2\times 7\times 7$

So, $\sqrt{784}=\sqrt{2\times 2\times 2\times 2\times 7\times 7}$

$=2\times 2\times 7$

$=28$

Hence, $\sqrt{784}=28$

(c) $4096$

Prime Factorisation:

$4096=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$

So, $\sqrt{4096}=\sqrt{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2}$

$=2\times 2\times 2\times 2\times 2\times 2$

$=8\times 8$

$=64$

Hence, $\sqrt{4096}=64$

(D) $7056$

Prime Factorisation:

$7056=2\times 2\times 2\times 2\times 3\times 3\times 7\times 7$

So, $\sqrt{7056}=\sqrt{2\times 2\times 2\times 2\times 3\times 3\times 7\times 7}$

$=2\times 2\times 3\times 7$

$=4\times 21$

$=84$

Hence, $\sqrt{7056}=84$