\(\sqrt{175} = \sqrt{25} ⋅ \sqrt{7} = 5 \sqrt{7} \text{.}\)

\(-6 \sqrt{27} = -6 ⋅ \sqrt{9} ⋅ \sqrt{3} = -6 ⋅ 3 ⋅ \sqrt{3} = -18 \sqrt{3} \text{.}\)

\(\sqrt{2\frac{1}{4}} = \sqrt{\frac{9}{4}} = {\sqrt{9} \over \sqrt{4}} = \frac{3}{2} = 1\frac{1}{2} \text{.}\)

\({24 \sqrt{400} \over 8 \sqrt{10}} = {24 \over 8} ⋅ {\sqrt{400} \over \sqrt{10}} = 3 \sqrt{40} = 3 ⋅ \sqrt{4} ⋅ \sqrt{10} = 3 ⋅ 2 ⋅ \sqrt{10} = 6 \sqrt{10}\)

\(\sqrt{1\frac{4}{9}} = \sqrt{\frac{13}{9}} = {\sqrt{13} \over \sqrt{9}} = {\sqrt{13} \over 3} = \frac{1}{3} \sqrt{13} \text{.}\)