\(\sqrt{500} + \sqrt{45} = \sqrt{100} ⋅ \sqrt{5} + \sqrt{9} ⋅ \sqrt{5} = 10 \sqrt{5} + 3 \sqrt{5} \text{.}\)

\(10 \sqrt{5} + 3 \sqrt{5} = 13 \sqrt{5} \text{.}\)

\(\sqrt{125} = \sqrt{25} ⋅ \sqrt{5} = 5 \sqrt{5} \text{.}\)

\(5 \sqrt{75} = 5 ⋅ \sqrt{25} ⋅ \sqrt{3} = 5 ⋅ 5 ⋅ \sqrt{3} = 25 \sqrt{3} \text{.}\)

\(5 \sqrt{8} + 2 \sqrt{32} = 5 ⋅ \sqrt{4} ⋅ \sqrt{2} + 2 ⋅ \sqrt{16} ⋅ \sqrt{2} \text{.}\)

\(5 ⋅ 2 ⋅ \sqrt{2} + 2 ⋅ 4 ⋅ \sqrt{2} = 10 \sqrt{2} + 8 \sqrt{2} = 18 \sqrt{2} \text{.}\)

\(\sqrt{\frac{25}{64}} = {\sqrt{25} \over \sqrt{64}} = \frac{5}{8} \text{.}\)