\( \begin{array}{ll}\text { a) } \frac{3^{x+1}-2 \cdot 3^{x}}{3^{x} \cdot 2} & \text { b) } \frac{2^{2 x+3}-5 \cdot 2^{2 x+1}}{4^{x+2}} \\ \text { c) } \frac{35^{a}-3 \cdot 5^{a}}{2^{2 a} \cdot 7 a \cdot 2^{2 a}} & \text { d) } \frac{3^{a+1} \cdot 4^{a}+5 \cdot 3^{a+1}}{4^{2 a}-25} \\ \text { e) } \frac{7^{a} \cdot 49-7^{a+2} \cdot 2^{-1}}{2^{-3} \cdot 7^{a}} & \text { f) } \frac{2^{3 a-1}+\frac{3}{2}}{2^{4 a-1}+3 \cdot 2^{a-1}} \\ \text { g) } \frac{-2^{3}-2^{5}}{5^{a+1} \cdot 2^{2}} & \text { h) } \frac{\frac{3}{4^{2}+4^{a-2}}}{3 \cdot 4^{3}+4^{a+3}}\end{array} \)

Imagine stepping back in time to the great mathematicians of ancient civilizations. The use of exponents can be traced back to the Egyptians and Babylonians, who used geometric principles and principles of pyramids that foreshadowed our understanding of exponential growth. They didn't have modern notation, but they sure had ideas that are foundational today! Now, when it comes to applying these algebraic expressions in the real world, think about how they appear in various fields such as finance for calculating compound interest, in physics for exponential decay or growth processes, and in computer science for algorithm analysis. Understanding how to manipulate these expressions can lead to powerful insights in data analysis, optimization, and more!