\[\sqrt[3]{1331}\]

\[=\sqrt[3]{11\times11\times11}\]

\[=\left(11^3\right)^{\frac{1}{3}}\]

\[=\left(11\right)^{3\times\frac{1}{3}}\]

\[=11\]

\[\sqrt{32}\]

\[=\sqrt{2\times2\times2\times2\times2}\]

\[=\sqrt{2^2\times2^2\times2}\]

\[=2^2\sqrt{2}\]

\[=4\sqrt{2}\]

\[\left(\frac{\sqrt{64}}{\sqrt{36}}\right)^2\]

\[=\left(\frac{\sqrt{8\times8}}{\sqrt{6\times6}}\right)^2\]

\[=\left(\frac{8}{6}\right)^2\]

\[=\frac{8^2}{6^2}\]

\[=\frac{64}{36}\]

\[=\frac{16}{9}\]

\[\left(\sqrt{5}\right)^2-\left(\sqrt{4}\right)^2\]

\[=5-4\]

\[=1\]

\[\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)\]

\[=\left(\sqrt{3}\times\sqrt{3}\right)+\left(\sqrt{3}\times-\sqrt{2}\right)+\left(\sqrt{2}\times\sqrt{3}\right)+\left(\sqrt{2}\times-\sqrt{2}\right)\]

\[=3-\sqrt{6}+\sqrt{6}-2\]

\[=1\]

\[\frac{\left(\sqrt{7}\right)^2-\left(\sqrt{5}\right)^2}{\sqrt{7}-\sqrt{5}}\]

\[=\frac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\sqrt{7}-\sqrt{5}}\]

\[=\sqrt{7}+\sqrt{5}\]

\[\sqrt{40+\sqrt{9\sqrt{81}}}\]

\[=\sqrt{40+\sqrt{9\times9}}\]

\[=\sqrt{40+9}\]

\[=\sqrt{49}\]

\[=7\]

\[\left(2\sqrt{3}+3\sqrt{2}\right)-\left(\sqrt{3}+\sqrt{2}\right)\]

\[=\left(2\sqrt{3}-\sqrt{3}\right)+\left(3\sqrt{2}-\sqrt{2}\right)\]

\[=\sqrt{3}+2\sqrt{2}\]

\[\sqrt{5+2\sqrt{6}}\]

\[=\sqrt{3+2+2\times\sqrt{3}\times\sqrt{2}}\]

\[=\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}\]

\[=\sqrt{\sqrt{3}+\sqrt{2}}\]

\[=\sqrt{3}+\sqrt{2}\]

\[(8)^{\frac{2}{3}}\]

\[=\left(2^3\right)^{\frac{2}{3}}\]

\[=2^{3\times\frac{2}{3}}\]

\[=2^2\]

\[=4\]

\[\left(\frac{1}{2}\right)^{-\frac{1}{2}}\]

\[=\left(2\right)^{\frac{1}{2}}\]

\[=\sqrt{2}\]

\[[3-4(3-4)^{-1}]^{-1}\]

\[=\left[3-4(-1)^{-1}\right]^{-1}\]

\[=\left[3-\frac{4}{-1}\right]^{-1}\]

\[=\left(3+4\right)^{-1}\]

\[=\left(7\right)^{-1}\]

\[=\frac{1}{7}\]

\[[(7.5)^2+2\times7.5\times2.5+(2.5)^2]\]

\[=[7.5+2.5]^2\]

\[=(10)^2\]

\[=100\]

\[3\sqrt{3}\times\sqrt{3}=3\times3=9.\]

\[\therefore\] Required rationalising factor is \[\sqrt{3}\].

Expression = \[\frac{1}{\sqrt{2}+1}\]

\[=\frac{\sqrt{2}-1}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}\]

\[=\frac{\sqrt{2}-1}{2-1}\]

\[=\sqrt{2}-1\]

\[=1.414-1\]

\[=0.414\]

\[\sqrt[3]{2}=2^{\frac{1}{3}}=2^{\frac{2}{6}}=\sqrt[6]{4}\]

\[\sqrt{3}=3^{\frac{1}{2}}=3^{\frac{2}{6}}=\sqrt[6]{27}\]

\[=\left(2\sqrt{2}-2-\sqrt{2}\right)\]

\[=\sqrt{2}-2\]

\[\left(\frac{8+2\sqrt{7}}{2}\right)\]

\[=\frac{8}{2}+\frac{2\sqrt{7}}{2}\]

\[=4+\sqrt{7}\]

\[\frac{1}{\sqrt{3}+\sqrt{4}}\]

\[=\frac{1}{\sqrt{3}+\sqrt{4}}\times\frac{\sqrt{4}-\sqrt{3}}{\sqrt{4}-\sqrt{3}}\]

\[=\frac{\sqrt{4}-\sqrt{3}}{\left(\sqrt{4}\right)^2-\left(\sqrt{3}\right)^2}\]

\[=\frac{\sqrt{4}-\sqrt{3}}{4-3}\]

\[=\frac{\sqrt{4}-\sqrt{3}}{1}\]

\[=\sqrt{4}-\sqrt{3}\]