1. Simplify $\frac{\sqrt[3]{m^{3}n^{6}} \times m^{4}n^{2}}{(\sqrt{m^{2}})^{2}n^{3}}$. Rewrite the cube root: $\sqrt[3]{m^{3}n^{6}} = m^{3/3}n^{6/3} = m^{1}n^{2}$. Multiply numerator: $m^{1}n^{2} \times m^{4}n^{2} = m^{1+4}n^{2+2} = m^{5}n^{4}$. Simplify denominator: $\sqrt{m^{2}} = m^{2/2} = m^{1}$, so $(\sqrt{m^{2}})^{2} = (m^{1})^{2} = m^{2}$. Denominator becomes $m^{2}n^{3}$. Divide numerator by denominator: $$\frac{m^{5}n^{4}}{m^{2}n^{3}} = m^{5-2}n^{4-3} = m^{3}n^{1} = m^{3}n$$ 2. Simplify $\frac{\sqrt{16xy} \times 7x^{3}y}{\sqrt{xy^{2}}}$. Simplify roots: $\sqrt{16xy} = 4\sqrt{xy}$ and $\sqrt{xy^{2}} = \sqrt{x}y$. Rewrite expression: $$\frac{4\sqrt{xy} \times 7x^{3}y}{\sqrt{x}y} = \frac{28x^{3}y\sqrt{xy}}{\sqrt{x}y}$$ Cancel $y$: $$\frac{28x^{3}\cancel{y}\sqrt{xy}}{\sqrt{x}\cancel{y}} = 28x^{3} \frac{\sqrt{xy}}{\sqrt{x}}$$ Simplify inside the root division: $$\frac{\sqrt{xy}}{\sqrt{x}} = \sqrt{\frac{xy}{x}} = \sqrt{y}$$ Final expression: $$28x^{3}\sqrt{y}$$ 3. Simplify $p^{1/2} (\sqrt[4]{q^{3}}) \times \sqrt[3]{p} (q^{-1})$. Rewrite roots: $$p^{1/2} = p^{0.5}, \quad \sqrt[4]{q^{3}} = q^{3/4}, \quad \sqrt[3]{p} = p^{1/3}$$ Multiply all terms: $$p^{0.5} \times q^{3/4} \times p^{1/3} \times q^{-1} = p^{0.5 + 1/3} q^{3/4 - 1}$$ Calculate exponents: $$0.5 + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$$ $$\frac{3}{4} - 1 = \frac{3}{4} - \frac{4}{4} = -\frac{1}{4}$$ Final expression: $$p^{5/6} q^{-1/4} = \frac{p^{5/6}}{q^{1/4}}$$ 4. Simplify $m^{8}\sqrt{n^{4}} \times \left(\frac{n}{m^{0}}\right)^{-2/3}$. Simplify root: $\sqrt{n^{4}} = n^{4/2} = n^{2}$. Since $m^{0} = 1$, the fraction is $\frac{n}{1} = n$. Apply exponent: $$\left(n\right)^{-2/3} = n^{-2/3}$$ Multiply all terms: $$m^{8} n^{2} \times n^{-2/3} = m^{8} n^{2 - 2/3}$$ Calculate exponent for $n$: $$2 - \frac{2}{3} = \frac{6}{3} - \frac{2}{3} = \frac{4}{3}$$ Final expression: $$m^{8} n^{4/3}$$