Let’s solve the given problems step by step.

Given Expressions:

1. a. $\frac{(-3)^7}{(-3)^5}$
2. b. $\frac{\left(\frac{5}{2}\right)^{10}}{\left(\frac{5}{2}\right)^4}$
3. c. $\frac{\left(\frac{y}{x}\right)^6}{\left(\frac{y}{x}\right)^4}$

Solutions:

Part a: Simplify $\frac{(-3)^7}{(-3)^5}$

Using the rule of exponents: $\frac{a^m}{a^n} = a^{m-n}$, $\frac{(-3)^7}{(-3)^5} = (-3)^{7-5} = (-3)^2.$ Final Answer: $(-3)^2$.

Part b: Simplify $\frac{\left(\frac{5}{2}\right)^{10}}{\left(\frac{5}{2}\right)^4}$

Again, using the rule of exponents: $\frac{a^m}{a^n} = a^{m-n}$, $\frac{\left(\frac{5}{2}\right)^{10}}{\left(\frac{5}{2}\right)^4} = \left(\frac{5}{2}\right)^{10-4} = \left(\frac{5}{2}\right)^6.$ Final Answer: $\left(\frac{5}{2}\right)^6$.

Part c: Simplify $\frac{\left(\frac{y}{x}\right)^6}{\left(\frac{y}{x}\right)^4}$

Using the same exponent rule: $\frac{\left(\frac{y}{x}\right)^6}{\left(\frac{y}{x}\right)^4} = \left(\frac{y}{x}\right)^{6-4} = \left(\frac{y}{x}\right)^2.$ Final Answer: $\left(\frac{y}{x}\right)^2$.

Final Answers:

• a. $(-3)^2$
• b. $\left(\frac{5}{2}\right)^6$
• c. $\left(\frac{y}{x}\right)^2$

Would you like me to further break down these solutions or provide additional examples?

Relative Questions:

1. What are the rules of exponents, and how can they be applied in different contexts?
2. How do negative bases affect the results of powers depending on whether the exponent is even or odd?
3. How can fractional exponents and base operations be combined for more complex simplifications?
4. What are the practical applications of these exponential simplifications in real-world problems?
5. How do these rules extend to logarithmic functions?

Tip:

Always simplify the numerator and denominator separately first when dealing with fractional bases before applying exponent rules.