1)
$\huge\frac{\left({a^\frac{3}{4}b^{-\frac{2}{3}}}\right)^{-\frac{1}{2}}}{\left({a^\frac{1}{2}b^{-\frac{2}{3}}}\right)^{-\frac{3}{4}}}$
$\huge\frac{\left({a^\frac{3}{4}b^{-\frac{2}{3}}}\right)^{-\frac{1}{2}}}{\left({a^\frac{1}{2}b^{-\frac{2}{3}}}\right)^{-\frac{3}{4}}}=$ $\huge\frac{a^{-\frac{3}{8}}b^\frac{1}{3}}{a^{-\frac{3}{8}}b^\frac{1}{2}}=b^{\frac{1}{3}-(\frac{1}{2})}=b^{\frac{2-3}{6}}=b^{-\frac{1}{6}}=\sqrt[6]{\frac{1}{b}}$
2)
$\huge\sqrt[5]{\left(\frac{\sqrt{a}\times a^{-1}}{\sqrt[3]{a}}\right)^{-3}}$
$\huge\sqrt[5]{\left(\frac{\sqrt{a}\times a^{-1}}{\sqrt[3]{a}}\right)^{-3}}=\left(\frac{a^{\frac{1}{2}}{a^{-1}}}{a^\frac{1}{3}}\right)^{-\frac{3}{5}}=\left(a^{\frac{3-6-2}{6}}\right)^{-\frac{3}{5}}\left(a^{-\frac{5}{6}}\right)^{-\frac{3}{5}}=a^\frac{1}{2}=\sqrt{a}$
3)
$\huge\sqrt{x\sqrt[3]{x}}\div \sqrt[3]{x\sqrt{x}}$
$\huge\sqrt{x\sqrt[3]{x}}\div \sqrt[3]{x\sqrt{x}}=x^{\frac{1}{2}} x^{\frac{1}{6}} \div x^{\frac{1}{3}} x^{\frac{1}{6}} = x^{\frac{1}{2}} \div x^{\frac{1}{3}} = x^{\frac{3-2}{6}} = x^{\frac{1}{6}} = \sqrt[6]{x}$
4)
$\huge\sqrt[3]{{\frac{a}{\sqrt{b^3}}}} \times \sqrt{{{\frac{b}{\sqrt[3]{a}}}}}$
$\huge\sqrt[3]{{\frac{a}{\sqrt{b^3}}}} \times \sqrt{{{\frac{b}{\sqrt[3]{a}}}}}=\frac{a^{\frac{1}{3}}}{b^{\frac{3}{6}}} \times \frac{b^{\frac{1}{2}}}{a^{\frac{1}{6}}}=a^{\frac{2-1}{6}}=a^{\frac{1}{6}}=\sqrt[6]{a}$
5)
$\huge \left[\left(a^3b\right)^\frac{1}{3}\right]^\frac{1}{2} \div \left[\left(a^3b^{-2}\right)^\frac{1}{2}\right]^\frac{1}{3}$
$\huge \left[\left(a^3b\right)^\frac{1}{3}\right]^\frac{1}{2} \div \left[\left(a^3b^{-2}\right)^\frac{1}{2}\right]^\frac{1}{3}=a^{\frac{1}{2}} b^{\frac{1}{6}} \div a^{\frac{1}{2}} b^{-\frac{1}{3}}=b^{\frac{1-(-2)}{6}}=b^{\frac{3}{6}} = \sqrt{b} $
6)
$\huge \frac{x\times x^\frac{1}{2}}{x^\frac{4}{5}\times\left(x^\frac{2}{3}\right)^\frac{1}{5}} \div \frac{\sqrt{x} \times x^{-1}}{\sqrt[3]{x}}=$
$\huge \frac{x\times x^\frac{1}{2}}{x^\frac{4}{5}\times\left(x^\frac{2}{3}\right)^\frac{1}{5}} \div \frac{\sqrt{x} \times x^{-1}}{\sqrt[3]{x}} =$ $\huge \frac{x^{\frac{2+1}{2}}}{x^{\frac{12+2}{15}}} \div \frac{x^{\frac{1-2}{2}}}{x^{\frac{1}{3}}}= \frac{x^{\frac{3}{2}}}{x^{\frac{14}{15}}} \times \frac{x^{\frac{1}{3}}}{x^{-\frac{1}{2}}}= \frac{x^{\frac{9+2}{6}}}{x^{\frac{28-15}{30}}}= \frac{x^{\frac{11}{6}}}{x^{\frac{13}{30}}}=x^{\frac{55-13}{30}}=x^{\frac{42}{30}}=x^{\frac{7}{5}}=\\ \huge \sqrt[5]{x^7}=\sqrt[5]{x^5x^2}=x\sqrt[5]{x^2} $
7)
$\huge \frac{a\sqrt{a}}{\sqrt[5]{a^4\sqrt[3]{a^2}}} \div \frac{a^\frac{1}{2}a^{-1}}{\sqrt[3]{a}}$
$\huge \frac{a\times a^{\frac{1}{2}}}{a^{\frac{4}{5}}a^{\frac{2}{15}}} \div \frac{a^\frac{1}{2}a^{-1}}{a^{\frac{1}{3}}}=$ $\huge \frac{a^{\frac{2+1}{2}}}{a^{\frac{12+2}{15}}} \div \frac{a^{\frac{1-2}{2}}}{a^{\frac{1}{3}}}= \frac{a^{\frac{3}{2}}}{a^{\frac{14}{15}}} \times \frac{a^{\frac{1}{3}}}{a^{-\frac{1}{2}}}= \frac{a^{\frac{9+2}{6}}}{a^{\frac{28-15}{30}}}= \frac{a^{\frac{11}{6}}}{a^{\frac{13}{30}}} \\ \huge=a^{\frac{55-13}{30}}=a^{\frac{42}{30}}=a^{\frac{7}{5}}=\sqrt[5]{a^{7}}=\sqrt[5]{a^5a^2}=a\sqrt[5]{a^2}$
8)
$\huge \sqrt{\frac{\sqrt[3]{xy^2}}{xy}} \times \sqrt[6]{x^2y}$
$\huge \sqrt{\frac{\sqrt[3]{xy^2}}{xy}} \times \sqrt[6]{x^2y}=\frac{x^{\frac{1}{6}}y^{\frac{2}{6}}}{x^{\frac{1}{2}}y^{\frac{1}{2}}} \times x^{\frac{2}{6}}y^{\frac{1}{6}} = \frac{x^{\frac{3}{6}}y^{\frac{3}{6}}}{x^{\frac{1}{2}}y^{\frac{1}{2}}} = 1$
9)
$\huge\sqrt{x\sqrt[3]{x^2}} + 4\sqrt[3]{x^2\sqrt{x}}$
$\huge\sqrt{x\sqrt[3]{x^2}} + 4\sqrt[3]{x^2\sqrt{x}}=x^{\frac{1}{2}}x^{{\frac{2}{6}}}+4x^{\frac{2}{3}}x^{\frac{1}{6}}=x^{\frac{3+2}{6}}+4x^{{\frac{4+1}{6}}}=x^{{\frac{5}{6}}}+4x^{{\frac{5}{6}}}=\\ \huge5x^{{\frac{5}{6}}}=5\sqrt[6]{x^{5}}$
10)
$\huge\frac{\sqrt[3]{x^{-2}}\times \sqrt{x^3}}{\sqrt[3]{\sqrt{x^4}}\times \sqrt{x^{-3}}}$
$\huge\frac{\sqrt[3]{x^{-2}}\times \sqrt{x^3}}{\sqrt[3]{\sqrt{x^4}}\times \sqrt{x^{-3}}}=$ $\huge \frac{x^{-\frac{2}{3}}x^{\frac{3}{2}}}{x^{\frac{4}{6}}x^{-\frac{3}{2}}}=$ $\huge\frac{x^{-\frac{-4+9}{6}}}{x^{\frac{4-9}{6}}}=$ $\huge\frac{x^{-\frac{5}{6}}}{x^{\frac{-5}{6}}}=x^{\frac{5-(-5)}{6}}=$ $\huge x^{\frac{10}{6}}=x^{\frac{5}{3}}=$ $\huge\sqrt[3]{x^{5}}=\sqrt[3]{x^{3}x^{2}}=x\sqrt[3]{x^{{2}}}$