\(\sqrt{8} = \sqrt{4} ⋅ \sqrt{2} = 2 \sqrt{2} \text{.}\)

\(4 \sqrt{12} = 4 ⋅ \sqrt{4} ⋅ \sqrt{3} = 4 ⋅ 2 ⋅ \sqrt{3} = 8 \sqrt{3} \text{.}\)

\(\sqrt{1\frac{24}{25}} = \sqrt{\frac{49}{25}} = {\sqrt{49} \over \sqrt{25}} = \frac{7}{5} = 1\frac{2}{5} \text{.}\)

\({18 \sqrt{160} \over 9 \sqrt{8}} = {18 \over 9} ⋅ {\sqrt{160} \over \sqrt{8}} = 2 \sqrt{20} = 2 ⋅ \sqrt{4} ⋅ \sqrt{5} = 2 ⋅ 2 ⋅ \sqrt{5} = 4 \sqrt{5}\)

\(\sqrt{\frac{71}{81}} = {\sqrt{71} \over \sqrt{81}} = {\sqrt{71} \over 9} = \frac{1}{9} \sqrt{71} \text{.}\)