\( \left. \begin{array} { l } { 7 ^ { \frac { 1 } { 3 } } * 7 ^ { \frac { 1 } { n } } = 49 ^ { \frac { 1 } { 7 } } } \\ { = \frac { 4 ^ { \frac { 1 } { 3 } } } { 20 ^ { 1 / 3 } } = 5 ^ { 1 / 3 } } \\ { \Rightarrow a ^ { m } + a ^ { n } = a ^ { m + n } } \\ { \Rightarrow a ^ { m } + a ^ { n } = a ^ { m \cdot n } } \end{array} \right. \)

Let's dive into the world of exponents! Did you know that the rules of exponents can be visualized as a fun game of combining powers? When you multiply bases, like \(7^{\frac{1}{3}}\) and \(7^{\frac{1}{n}}\), you're essentially stacking, which leads to \(7^{(\frac{1}{3} + \frac{1}{n})}\). This magic works because multiplying the same base allows you to add the exponents. Just remember, \(a^m \times a^n = a^{m+n}\)! Now, if you're looking to master exponents, a great real-world application is in finance! Compound interest formulas use exponents to predict how money grows over time. For example, if you invest money that compounds annually, the formula \(A = P(1 + r)^t\) uses exponents to show how your investment \(P\) increases at rate \(r\) over time \(t\). This can help you strategize your investments and grow your wealth!