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Let's work through these simplifications step by step. We will apply the rules of exponents to simplify each expression:

### 1. \(\left(\frac{b}{h}\right)^3\)
Apply the power to both the numerator and denominator:
\[
\frac{b^3}{h^3}
\]

### 2. \(\left(\frac{2}{c^2}\right)^2\)
Distribute the exponent of 2 to both the numerator and denominator:
\[
\frac{2^2}{c^{2 \times 2}} = \frac{4}{c^4}
\]

### 3. \(\left(\frac{rq}{ds}\right)^5\)
Distribute the exponent of 5 to each term in the numerator and denominator:
\[
\frac{r^5 q^5}{d^5 s^5}
\]

### 4. \(\left(\frac{ab}{cd}\right)^6\)
Distribute the exponent of 6:
\[
\frac{a^6 b^6}{c^6 d^6}
\]

### 5. \(\left(\frac{abc}{xyz}\right)^8\)
Distribute the exponent of 8:
\[
\frac{a^8 b^8 c^8}{x^8 y^8 z^8}
\]

### 6. \(\left(\frac{x^2 y^3 z}{a^5 b^4 c^2}\right)^6\)
Distribute the exponent of 6 to each term:
\[
\frac{x^{12} y^{18} z^6}{a^{30} b^{24} c^{12}}
\]

### 7. \(\left(\frac{m^n n}{n^2}\right)^4\)
Simplify inside the parentheses first:
\[
\frac{m^n n}{n^2} = \frac{m^n}{n}
\]
Now apply the exponent of 4:
\[
\frac{m^{4n}}{n^4}
\]

### 8. \(\left(\frac{a^3 b^2}{c^4}\right)^5\)
Distribute the exponent of 5:
\[
\frac{a^{15} b^{10}}{c^{20}}
\]

### 9. \(\left(\frac{c^2 d^3}{a^4 b^5}\right)^7\)
Distribute the exponent of 7:
\[
\frac{c^{14} d^{21}}{a^{28} b^{35}}
\]

### 10. \(\left(\frac{2m^2 n^3}{3m^4 n}\right)^2\)
Simplify inside the parentheses first:
\[
\frac{2m^2 n^3}{3m^4 n} = \frac{2n^2}{3m^2}
\]
Now apply the exponent of 2:
\[
\frac{(2n^2)^2}{(3m^2)^2} = \frac{4n^4}{9m^4}
\]

### 11. \(\left(\frac{7ab^3}{8c^2 d^5}\right)^2\)
Distribute the exponent of 2:
\[
\frac{49a^2 b^6}{64c^4 d^{10}}
\]

### 12. \(\left(\frac{5m^2 n^4}{6ab^2 c}\right)^3\)
Distribute the exponent of 3:
\[
\frac{125m^6 n^{12}}{216a^3 b^6 c^3}
\]

### 13. \(\left(\frac{-x}{4}\right)^2\)
Apply the exponent of 2:
\[
\frac{x^2}{16}
\]

### 14. \(\left(\frac{10x^5}{2x^2}\right)^2\)
Simplify inside the parentheses:
\[
\frac{10x^5}{2x^2} = 5x^3
\]
Now apply the exponent of 2:
\[
25x^6
\]

### 15. \(\left(\frac{a^3}{b^6}\right)^5\)
Distribute the exponent of 5:
\[
\frac{a^{15}}{b^{30}}
\]

### 16. \(\left(\frac{2a^2 b^3 c}{3x^3 y^2 z^3}\right)^3\)
Distribute the exponent of 3:
\[
\frac{8a^6 b^9 c^3}{27x^9 y^6 z^9}
\]

### 17. \(\left(\frac{3a^2 n^2}{4x^2 n^3}\right)^2\)
Simplify inside the parentheses:
\[
\frac{3a^2 n^2}{4x^2 n^3} = \frac{3a^2}{4x^2 n}
\]
Now apply the exponent of 2:
\[
\frac{9a^4}{16x^4 n^2}
\]

### 18. \(\left(\frac{ab^2}{2x^3}\right)^5\)
Distribute the exponent of 5:
\[
\frac{a^5 b^{10}}{32x^{15}}
\]

### 19. \(\left(\frac{x^4 y^5}{x^3 y^2}\right)^7\)
Simplify inside the parentheses:
\[
\frac{x^4 y^5}{x^3 y^2} = x y^3
\]
Now apply the exponent of 7:
\[
x^7 y^{21}
\]

### 20. \(\left(\frac{2a}{3b^2}\right)^3\)
Distribute the exponent of 3:
\[
\frac{8a^3}{27b^6}
\]

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If you have any questions or need more details, feel free to ask! Here are five questions related to this problem:

1. How do you simplify expressions with negative exponents?
2. What happens if the base inside the parentheses is negative and the exponent is even?
3. How can you use the power of a quotient rule when the exponents are fractional?
4. How do you simplify expressions involving zero exponents?
5. How do you distribute exponents when there are multiple variables in the expression?

**Tip:** Always simplify inside the parentheses first before applying the exponent outside the parentheses for easier calculation.

Simplify the following expressions. Assume all denominators are not equal to zero.

Learn how to simplify expressions with rational quotients and exponents. This problem set covers key algebraic rules, including the power of a quotient rule, distributive property of exponents, and exponent multiplication. Ideal for students in Grades 9-12.

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