Examples of Solving Powers with Variables

$$\begin{array}{llll}\hfill {(xy)}^3\cdot x^2{y}^{-3}& =x^3{y}^3{x}^2{y}^{-3}& \hfill & \\ \hfill & =x^{3+2}{y}^{3-3}& \hfill & \\ \hfill & =x^5{y}^0& \hfill & \\ \hfill & =x^5\cdot 1& \hfill & \\ \hfill & =x^5& \hfill & \end{array}$$

$${\left(\frac{x}{y}\right)}^{-3}=\frac{x^{-3}}{y^{-3}}=\frac{1}{x^3{y}^{-3}}=\frac{y^3}{x^3}$$

Note that you can actually just flip the fraction and change the sign of the exponent, like this:

$${\left(\frac{x}{y}\right)}^{-3}={\left(\frac{y}{x}\right)}^3=\frac{y^3}{x^3}$$

$$\begin{array}{llll}\hfill 3^3{a}^{-3}b{(ab)}^3& =27a^{-3}b{a}^3{b}^3& \hfill & \\ \hfill & =27a^{-3+3}{b}^{1+3}& \hfill & \\ \hfill & =27a^0{b}^4& \hfill & \\ \hfill & =27b^4& \hfill & \end{array}$$

$$\begin{array}{llll}\hfill \frac{2x^2{y}^2}{2xy}& =\frac{2^1{x}^2{y}^2}{2^1{x}^1{y}^1}& \hfill & \\ \hfill & =\frac{2^1{2}^{-1}{x}^2{x}^{-1}{y}^2{y}^{-1}}{1}& \hfill & \\ \hfill & =2^{1-1}{x}^{2-1}{y}^{2-1}& \hfill & \\ \hfill & =2^0{x}^1{y}^1& \hfill & \\ \hfill & =xy& \hfill & \end{array}$$

$$\begin{array}{llll}\hfill \frac{a^2{b}^{-4}{2}^2}{{(2ab)}^{-3}}& =\frac{a^2{b}^{-4}{2}^2}{2^{-3}{a}^{-3}{b}^{-3}}& \hfill & \\ \hfill & =a^2{a}^3{b}^{-4}{b}^3{2}^2{2}^3& \hfill & \\ \hfill & =a^{2+3}{b}^{-4+3}{2}^{2+3}& \hfill & \\ \hfill & =a^5{b}^{-1}{2}^5& \hfill & \\ \hfill & =\frac{2^5{a}^5}{b}& \hfill & \end{array}$$