To simplify, we must try to find factors which are perfect squares. In this case 16 is a factor of 624 and is also a perfect square.

Therefore we can rewrite the square root of 624 as:

\(\displaystyle \sqrt{624}=\sqrt{16\times 39}=\sqrt{16}\sqrt{39}=4\sqrt{39}\)

To simplify, we must try to find factors which are perfect squares. In this case 20 is a factor of 400 and is also a perfect square.

Thus we can rewrite the problem as:

\(\displaystyle \sqrt{400}=\sqrt{20\times 20}=20\)

Note: \(\displaystyle 20^{2}=20\times20=400\)

Use the following steps to reduce this square root.

To simplify, we must try to find factors which are perfect squares. In this case 144 is a factor of 720 and is also a perfect square.

Thus we can rewrite the problem as follows.

\(\displaystyle \sqrt{720}=\sqrt{144\times 5}=\sqrt{144}\sqrt{5}=12\sqrt{5}\)

Use the following steps to find the square root of \(\displaystyle 1,800:\)

To simplify, we must try to find factors which are perfect squares. In this case 900 is a factor of 1800 and is also a perfect square.

Thus we can rewrite the problem as follows.

\(\displaystyle \sqrt{1,800}=\sqrt{900\times 2}=\sqrt{900}\sqrt{2}=30\sqrt{2}\)

To simplify, we must try to find factors which are perfect squares. In this case 9 is a factor of 54 and is also a perfect square.

To reduce this expression, use the following steps: 

\(\displaystyle \sqrt{54}=\sqrt{9\times 6}=\sqrt{9}\sqrt{6}=3\sqrt{6}\)

To simplify, we must try to find factors which are perfect squares. In this case 36 is a factor of 72 and is also a perfect square.

To reduce this expression, use the following arithmetic steps: 

\(\displaystyle \sqrt{72}=\sqrt{36\times 2}=\sqrt{36}\sqrt{2}=6\sqrt{2}\)

To simplify, we must try to find factors which are perfect squares. In this case 30 is a factor of 900 and is also a perfect square.

The square root of \(\displaystyle 900\) is equal to: 

\(\displaystyle \sqrt{900}=\sqrt{30\times 30}=30\)

However,

\(\displaystyle \sqrt{900}=\sqrt{-30\times -30}=-30\)

Thus, \(\displaystyle \pm30\) \(\displaystyle =\sqrt{900}\)

To simplify, we must try to find factors which are perfect squares. In this case 16 is a factor of 32 and is also a perfect square.

To reduce this expression, use the following steps:

\(\displaystyle \sqrt{32}=\sqrt{16\times 2}=\sqrt{16}\sqrt{2}=4\sqrt{2}\)

To simplify, we must try to find factors which are perfect squares. In this case 4 is a factor of 164 and is also a perfect square.

To find the square root of \(\displaystyle 164\), use the following steps:

\(\displaystyle \sqrt{164}=\sqrt{4\times 41}=\sqrt{41}\sqrt{4}=2\sqrt{41}\)

Use the following arithmetic steps to reduce \(\displaystyle \sqrt{192}\).

To simplify, we must try to find factors which are perfect squares. In this case 64 is a factor of 192 and is also a perfect square.

Note \(\displaystyle 64\) and \(\displaystyle 3\) are both factors of \(\displaystyle 192\), however only \(\displaystyle \sqrt{64}\) can be reduced.


\(\displaystyle \sqrt{192}=\sqrt{64\times 3}=\sqrt{64}\sqrt{3}=8\sqrt{3}\)