Samiyah Sampson Use the relevant properties of exponents to simplify the following expression. Answers should only have positive exponents. \begin{tabular}{|l|l|}\hline \( 7 \wedge 72 \) & \( \left(7^{8}\right)^{-9} \) \\ \hline\end{tabular}

When simplifying \( 7^{72} \), it stays as is since it’s already in its simplest form with positive exponents. Moving on to the second expression, \( \left(7^{8}\right)^{-9} \) can be simplified using the property of exponents that states \( (a^{m})^{n} = a^{m \cdot n} \). This gives us: \[ (7^{8})^{-9} = 7^{8 \cdot (-9)} = 7^{-72} \] To express it with positive exponents, we can rewrite it as: \[ \frac{1}{7^{72}} \] So, the simplified forms of the expressions are \( 7^{72} \) and \( \frac{1}{7^{72}} \). Now, here's a fun fact: exponents have been used since ancient civilizations, particularly by the Babylonians! They employed various methods to handle numbers of large magnitude without the notational systems we're familiar with today. Additionally, understanding how to manipulate exponents is essential not just in math class but also in real life. For example, they're used in finance to calculate compound interest, where the formula involves exponents exponentially changing your investments!