\(\sqrt{75} + \sqrt{27} = \sqrt{25} ⋅ \sqrt{3} + \sqrt{9} ⋅ \sqrt{3} = 5 \sqrt{3} + 3 \sqrt{3} \text{.}\)

\(5 \sqrt{3} + 3 \sqrt{3} = 8 \sqrt{3} \text{.}\)

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

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

\(3 \sqrt{27} - 6 \sqrt{12} = 3 ⋅ \sqrt{9} ⋅ \sqrt{3} - 6 ⋅ \sqrt{4} ⋅ \sqrt{3} \text{.}\)

\(3 ⋅ 3 ⋅ \sqrt{3} - 6 ⋅ 2 ⋅ \sqrt{3} = 9 \sqrt{3} - 12 \sqrt{3} = -3 \sqrt{3} \text{.}\)

\(\sqrt{1\frac{19}{81}} = \sqrt{\frac{100}{81}} = {\sqrt{100} \over \sqrt{81}} = \frac{10}{9} = 1\frac{1}{9} \text{.}\)