Calculation rules of powers

\[\begin{array}{||c|c|c||} \hline \textit{Numeric example} & \textit{Formula example} & \textit{General rule}\\ \hline 5^{\frac{3}{2}}\times 5^{\frac{1}{2}}=5^2 & x^{\frac{3}{2}}\cdot x^{\frac{1}{2}}=x^2 & x^r\cdot x^s = x^{r+s}\\[10pt] \dfrac{3^2}{3^{\frac{5}{2}}}=3^{-\frac{1}{2}} & \dfrac{x^2}{x^{\frac{5}{2}}}=x^{-\frac{1}{2}} & \dfrac{x^r}{x^s}= x^{r-s} \\[10pt] (2^{\frac{1}{2}})^4=2^{2} & (x^{\frac{1}{2}})^4=x^{2} & (x^r)^s=x^{r\cdot s}\\[10pt] (2\times 3)^{\frac{1}{4}}=2^{\frac{1}{4}}\times 3^{\frac{1}{4}} & (xy)^4 = x^4y^4 & (x\cdot y)^r= x^r\cdot y^r\\[10pt] \left(\dfrac{2}{3}\right)^4=\dfrac{2^4}{3^4} & \left(\dfrac{x}{y}\right)^{4}=\dfrac{x^{4}}{y^{4}} & \left(\dfrac{x}{y}\right)^r=\dfrac{x^r}{y^r} \\[10pt]\hline\end{array}\]
for variables \(x\) and \(y\) and for all rational numbers \(r\) and \(s\).