Question

Find the square root of $1728$ in radical form

Answer 1

Solve for the square root of given number:

Given number is $1728$

Writing $1728$ as product of prime numbers, we get,

$$\begin{gathered} 1728=2\times 864 \\ =2\times 2\times 432 \\ =2\times 2\times 2\times 216 \\ =2\times 2\times 2\times 2\times 108 \\ =2\times 2\times 2\times 2\times 2\times 54 \\ =2\times 2\times 2\times 2\times 2\times 2\times 27 \\ =2\times 2\times 2\times 2\times 2\times 2\times 3\times 9 \\ =2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3 \end{gathered}$$

$\therefore 1728=2\times 2\times 3\times 3\times 3\times 2\times 2\times 2\times 2$

Now we have all the prime numbers with us. Taking the root on both sides,

$\Rightarrow \sqrt{1728}=\sqrt{2\times 2\times 3\times 3\times 3\times 2\times 2\times 2\times 2}$

$$\begin{gathered} =2\times 2\times 2\times 3\times \sqrt{3} \\ \therefore \sqrt{1728}=24\sqrt{3} \end{gathered}$$

Hence, the square root of $1728$ in radical form is $24\sqrt{3}$.