\(4{p^4}{q^4}\)

$$
\begin{aligned}
\frac{(4 p)^{2} \times 2 q^{3}}{8 p^{-2} q^{-1}}&=\frac{16 p^{2} \times 2 q^{3}}{8 p^{-2} q^{-1}}\\ &=\frac{32 p^{2} q^{3}}{8 p^{-2} q^{-1}} \\
&=4 p^{2} q^{3} \times p^{2} q^{1} \\
&=4 p^{4} q^{4}
\end{aligned}
$$

\(a = 1,\,\,b = -5\)

$$
\begin{aligned}
{\frac{3^{n} \times 9^{n-1}}{3^{2 n} \times 27}}&=\frac{3^{n} \times\left(3^{2}\right)^{n-1}}{3^{2 n} \times 3^{3}}\\
&=\frac{3^{n} \times 3^{2 n-2}}{3^{2 n+3}} \\
&=3^{3 n-2} \times 3^{-2 n-3} \\
&=3^{n-5} \\
& \therefore a=1,b=-5
\end{aligned}
$$

\({2^{\frac{2}{3}}}\)

$$
\begin{aligned}
\left(2^{\frac{1}{4}}\right)^{2} \times 2^{\frac{1}{3}} \times 2^{-\frac{1}{6}} &=2^{\frac{1}{2}} \times 2^{\frac{1}{3}} \times 2^{-\frac{1}{6}} \\
&=2^{\frac{1}{2}+\frac{1}{3}-\frac{1}{6}} \\
&=2^{\frac{2}{3}}
\end{aligned}
$$

\(x{y^3}\)

$$
\begin{aligned}
\sqrt{x^{3} y^{2}} \times \sqrt{x^{-1} y^{4}} &=\sqrt{x^{3} y^{2} \times x^{-1} y^{4}} \\
&=\left(x^{2} y^{6}\right)^{\frac{1}{2}} \\
&=x y^{3}
\end{aligned}
$$