[tex](\frac{13^0}{13^3})^{\frac{1}{3}}=13^{-\frac{2}{3}}\text{ is not equivalent}[/tex]Explanation:[tex]\begin{gathered} \text{Given:} \\ (\frac{13^0}{13^3})^{\frac{1}{3}}=13^{-\frac{2}{3}} \\ (\frac{13^{}}{13^3})^{\frac{1}{3}}=(13^{-2})^{\frac{1}{3}} \end{gathered}[/tex]
To determine why the two expressions/statements are wrong, we need to solve each seperately and compare the result
[tex]\begin{gathered} (\frac{13^0}{13^3})^{\frac{1}{3}}=13^{-\frac{2}{3}} \\ \text{left hand side of the equation: }(\frac{13^0}{13^3})^{\frac{1}{3}} \\ \text{The base of the left side is common, we simplify by subtracting the exponents:} \\ (13^{0-3})^{\frac{1}{3}}=(13^{-3})^{\frac{1}{3}} \\ =13^{-\frac{3}{3}\text{ }}=13^{-1} \\ \\ \text{The result of the left hand side is not the }sa\text{me as the right hand side} \\ 13^{-1}\text{ }\ne\text{ }13^{-\frac{2}{3}} \end{gathered}[/tex][tex]\begin{gathered} (\frac{13^{}}{13^3})^{\frac{1}{3}}=(13^{-2})^{\frac{1}{3}} \\ \text{left hand side: }(\frac{13^{}}{13^3})^{\frac{1}{3}} \\ (\frac{13^{}}{13^3})^{\frac{1}{3}}=(\frac{13^1^{}}{13^3})^{\frac{1}{3}} \\ \text{The base are common, subtract exponents as they are seperated by division:} \\ (13^{1-3})^{\frac{1}{3}}=(13^{-2})^{\frac{1}{3}} \\ =13^{-\frac{2}{3}} \\ \\ \text{right hand side:} \\ (13^{-2})^{\frac{1}{3}}=13^{-\frac{2}{3}} \\ \\ \text{The left hand side = right hand side} \\ \text{Hence, }(\frac{13^{}}{13^3})^{\frac{1}{3}}=(13^{-2})^{\frac{1}{3}} \end{gathered}[/tex]
From our workings, we see the first equation statement is wrong so we cannot compare it with the other equation for equivalence
[tex]\text{Hence, }(\frac{13^0}{13^3})^{\frac{1}{3}}=13^{-\frac{2}{3}}\text{ is not equivalent}[/tex]
