(1)) Simplify. Assume all variables are positive. \[ u^{\frac{7}{3}} \div u^{\frac{8}{3}} \] Write your answer in the form \( A \) or \( \frac{A}{B^{\prime}} \) where \( A \) and \( B \) are

To simplify \( u^{\frac{7}{3}} \div u^{\frac{8}{3}} \), we use the properties of exponents which state that when dividing like bases, we subtract the exponents: \[ u^{\frac{7}{3} - \frac{8}{3}} = u^{-\frac{1}{3}}. \] Now, to express it in the required form, we can rewrite \( u^{-\frac{1}{3}} \) as: \[ \frac{1}{u^{\frac{1}{3}}}. \] So, the simplified expression is: \[ \frac{1}{u^{\frac{1}{3}}}. \] --- In ancient Egypt, the use of exponents began around 2000 BC in mathematics, particularly in calculating areas and volumes. Ancient mathematicians used these principles to construct impressive monuments and aid in trade, laying the groundwork for modern algebra. In today's world, exponents are critical in fields such as computer science, engineering, and finance. For instance, efficient algorithms often rely on logarithmic and exponential functions to process data rapidly, making the understanding of exponents paramount in technology-driven industries!