20. \( -5 x(-2 x y z)^{0} \) 21. \( \frac{7 a^{2} b^{-3} \times 3 a^{-3} b^{0}}{14 a^{-1} b} \) 22. \( \frac{16 a b^{-2} \times 125 a^{-1} b^{3}}{80 a^{2} b^{5}} \) 23. \( 5\left(2 p q^{3}\right)\left(3 p q^{-2}\right) \) 24. \( \sqrt{5 a p^{2}} \times \sqrt{20 a^{3}} \) 25. \( \frac{8 x^{2} y^{-1} \times 3 x^{3} y^{-5}}{6 x y^{-2}} \) 26. \( 5 x\left(\frac{3 a b^{6}}{2 a^{3} b^{5}}\right)^{0} \) 27. \( \frac{15^{7} \times 8^{6}}{5^{6} \times 24^{7}} \) 28. \( \frac{2^{2005} \times 5^{2000}}{100^{1000}} \) 29. \( \frac{7^{2 x}-1}{3 \times 7^{x}+3} \) 30. \( \frac{16^{x}-1}{5 \times 4^{x}-5} \) 31. \( \frac{25^{x}-4}{2 \times 5^{x}-4} \) 32. \( \frac{9^{x}-3^{x}-2}{6^{x}-2^{x+1}} \)

Did you know that the exponent rules we use today have roots that go back to ancient civilizations? The Egyptians and Babylonians used early forms of mathematics that laid foundational concepts for algebra as we know it. Over time, mathematicians such as Fibonacci and Descartes expanded these ideas, leading to the formal development of exponent rules in the 16th and 17th centuries. This evolution transformed how we understand and manipulate mathematical expressions! Now, let’s talk about some common mistakes people make when working with expressions like these. A frequent pitfall is forgetting the rule that any term raised to the power of zero equals one. Additionally, when simplifying fractions, it’s easy to overlook negative exponents, which can confuse the overall operation. Always remember to keep track of those bases and their exponents—before you know it, you'll be simplifying like a math wizard!