Practice Test-Simplify, Multiply, Divide, Solve Equations with Rational Expressions 1-20-2023 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. $$ \frac{4 x-8}{4 x+20} $$ a. $$ \frac{x-2}{4 x+20} ; $$ where $$ x
eq-5 $$ c. $$ \frac{x-2}{x+5} $$; where $$ x
eq-5 $$ b. $$ \frac{x+5}{x-2} $$; where $$ x
eq 2 $$ d. $$ \quad 4\left(\frac{x+2}{x-5} ight) ; $$ where $$ x
eq 5 $$ 2. $$ \frac{x^{2}-64}{80 x-10 x^{2}} $$ a. $$ \frac{x-8}{10 x} $$; where $$ x
eq 0,-8 $$ c. $$ \quad-\frac{x+8}{10 x} $$; where $$ x
eq 0,8 $$ b. $$ \frac{8-x}{10 x} $$; where $$ x
eq 0,-8 $$ d. $$ \frac{8-x}{10 x} $$; where $$ x
eq 0,8 $$ 3. $$ \frac{4 a^{5}}{7 b^{4}} \cdot \frac{2 b^{2}}{2 a^{4}} $$ a. $$ \frac{4 a^{9}}{7 b^{6}}, a
eq 0, b
eq 0 $$ c. $$ \frac{7 b^{2}}{4 a}, a
eq 0, b
eq 0 $$ b. $$ \frac{4 a}{7 b^{2}}, a
eq 0, b
eq 0 $$ d. $$ \frac{4}{7} a^{9} b^{6}, a
eq 0, b
eq 0 $$ 4. $$ \frac{a^{2}}{a-3} \cdot \frac{a^{2}-2 a-3}{a^{2}-6 a} $$ a. $$ \frac{a^{2}+a}{a-6}, a
eq 3,0,6 $$ c. $$ \frac{a+1}{a-6}, a
eq 3,0,6 $$ b. $$ \frac{a^{2}+a}{a-6}, a
eq 3,6 $$ d. $$ \frac{a+1}{a-6}, a
eq 3,6 $$ 5. $$ \frac{\frac{n-6}{n^{2}+11 n+24}}{\frac{n+1}{n+3}} $$ a. $$ \frac{(n-6)(n+1)}{(n+3)^{2}(n+8)} $$ c. $$ \frac{(n-6)(n+1)}{(n+3)(n+8)} $$ b. $$ \frac{(n-6)(n+8)}{(n+1)(n-8)} $$ d. $$ \frac{n-6}{(n+1)(n+8)} $$

Question

Practice Test-Simplify, Multiply, Divide, Solve Equations with Rational Expressions 1-20-2023 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. $$ \frac{4 x-8}{4 x+20} $$ a. $$ \frac{x-2}{4 x+20} ; $$ where $$ x 
eq-5 $$ c. $$ \frac{x-2}{x+5} $$; where $$ x 
eq-5 $$ b. $$ \frac{x+5}{x-2} $$; where $$ x 
eq 2 $$ d. $$ \quad 4\left(\frac{x+2}{x-5}ight) ; $$ where $$ x 
eq 5 $$ 2. $$ \frac{x^{2}-64}{80 x-10 x^{2}} $$ a. $$ \frac{x-8}{10 x} $$; where $$ x 
eq 0,-8 $$ c. $$ \quad-\frac{x+8}{10 x} $$; where $$ x 
eq 0,8 $$ b. $$ \frac{8-x}{10 x} $$; where $$ x 
eq 0,-8 $$ d. $$ \frac{8-x}{10 x} $$; where $$ x 
eq 0,8 $$ 3. $$ \frac{4 a^{5}}{7 b^{4}} \cdot \frac{2 b^{2}}{2 a^{4}} $$ a. $$ \frac{4 a^{9}}{7 b^{6}}, a 
eq 0, b 
eq 0 $$ c. $$ \frac{7 b^{2}}{4 a}, a 
eq 0, b 
eq 0 $$ b. $$ \frac{4 a}{7 b^{2}}, a 
eq 0, b 
eq 0 $$ d. $$ \frac{4}{7} a^{9} b^{6}, a 
eq 0, b 
eq 0 $$ 4. $$ \frac{a^{2}}{a-3} \cdot \frac{a^{2}-2 a-3}{a^{2}-6 a} $$ a. $$ \frac{a^{2}+a}{a-6}, a 
eq 3,0,6 $$ c. $$ \frac{a+1}{a-6}, a 
eq 3,0,6 $$ b. $$ \frac{a^{2}+a}{a-6}, a 
eq 3,6 $$ d. $$ \frac{a+1}{a-6}, a 
eq 3,6 $$ 5. $$ \frac{\frac{n-6}{n^{2}+11 n+24}}{\frac{n+1}{n+3}} $$ a. $$ \frac{(n-6)(n+1)}{(n+3)^{2}(n+8)} $$ c. $$ \frac{(n-6)(n+1)}{(n+3)(n+8)} $$ b. $$ \frac{(n-6)(n+8)}{(n+1)(n-8)} $$ d. $$ \frac{n-6}{(n+1)(n+8)} $$

UpStudy AI · Accepted Answer

Let's solve each of the problems step by step.

### Problem 1: Simplify $$ \frac{4x-8}{4x+20} $$

1. **Factor the numerator and denominator:**
   - The numerator $$ 4x - 8 $$ can be factored as $$ 4(x - 2) $$.
   - The denominator $$ 4x + 20 $$ can be factored as $$ 4(x + 5) $$.

2. **Rewrite the expression:**
   $$
   \frac{4(x - 2)}{4(x + 5)}
   $$

3. **Cancel the common factor of 4:**
   $$
   \frac{x - 2}{x + 5}
   $$

4. **Final expression:**
   $$
   \frac{x - 2}{x + 5}, \quad x \neq -5
   $$

**Answer:** c. $$ \frac{x-2}{x+5} $$; where $$ x \neq -5 $$

---

### Problem 2: Simplify $$ \frac{x^{2}-64}{80x-10x^{2}} $$

1. **Factor the numerator and denominator:**
   - The numerator $$ x^2 - 64 $$ is a difference of squares: $$ (x - 8)(x + 8) $$.
   - The denominator $$ 80x - 10x^2 $$ can be factored as $$ -10x^2 + 80x = -10x(x - 8) $$.

2. **Rewrite the expression:**
   $$
   \frac{(x - 8)(x + 8)}{-10x(x - 8)}
   $$

3. **Cancel the common factor $$ (x - 8) $$:**
   $$
   \frac{x + 8}{-10x}
   $$

4. **Final expression:**
   $$
   -\frac{x + 8}{10x}, \quad x \neq 0, 8
   $$

**Answer:** c. $$ -\frac{x+8}{10x} $$; where $$ x \neq 0, 8 $$

---

### Problem 3: Multiply $$ \frac{4a^{5}}{7b^{4}} \cdot \frac{2b^{2}}{2a^{4}} $$

1. **Multiply the numerators and denominators:**
   $$
   \frac{4a^{5} \cdot 2b^{2}}{7b^{4} \cdot 2a^{4}} = \frac{8a^{5}b^{2}}{14b^{4}a^{4}}
   $$

2. **Simplify the expression:**
   - Cancel $$ 2 $$ from numerator and denominator: 
   $$
   \frac{4a^{5}b^{2}}{7b^{4}a^{4}}
   $$
   - Cancel $$ a^{4} $$ from $$ a^{5} $$ and $$ b^{2} $$ from $$ b^{4} $$:
   $$
   \frac{4a^{1}}{7b^{2}}
   $$

3. **Final expression:**
   $$
   \frac{4a}{7b^{2}}, \quad a \neq 0, b \neq 0
   $$

**Answer:** b. $$ \frac{4a}{7b^{2}}, a \neq 0, b \neq 0 $$

---

### Problem 4: Multiply $$ \frac{a^{2}}{a-3} \cdot \frac{a^{2}-2a-3}{a^{2}-6a} $$

1. **Factor the second numerator and denominator:**
   - The numerator $$ a^2 - 2a - 3 $$ factors to $$ (a - 3)(a + 1) $$.
   - The denominator $$ a^2 - 6a $$ factors to $$ a(a - 6) $$.

2. **Rewrite the expression:**
   $$
   \frac{a^{2}}{a-3} \cdot \frac{(a - 3)(a + 1)}{a(a - 6)}
   $$

3. **Cancel the common factor $$ (a - 3) $$:**
   $$
   \frac{a(a + 1)}{a(a - 6)}
   $$

4. **Cancel $$ a $$ (assuming $$ a \neq 0 $$):**
   $$
   \frac{a + 1}{a - 6}, \quad a \neq 3, 0, 6
   $$

**Answer:** d. $$ \frac{a+1}{a-6}, a \neq 3, 6 $$

---

### Problem 5: Simplify $$ \frac{\frac{n-6}{n^{2}+11n+24}}{\frac{n+1}{n+3}} $$

1. **Factor the denominator of the first fraction:**
   - The denominator $$ n^2 + 11n + 24 $$ factors to $$ (n + 3)(n + 8) $$.

2. **Rewrite the expression:**
   $$
   \frac{n - 6}{(n + 3)(n + 8)} \cdot \frac{n + 3}{n + 1}
   $$

3. **Cancel the common factor $$ (n + 3) $$:**
   $$
   \frac{(n - 6)}{(n + 8)(n + 1)}
   $$

4. **Final expression:**
   $$
   \frac{n - 6}{(n + 1)(n + 8)}
   $$

**Answer:** d. $$ \frac{n-6}{(n+1)(n+8)} $$

---

### Summary of Answers:
1. c. $$ \frac{x-2}{x+5} $$; where $$ x \neq -5 $$
2. c. $$ -\frac{x+8}{10x} $$; where $$ x \neq 0, 8 $$
3. b. $$ \frac{4a}{7b^{2}}, a \neq 0, b \neq 0 $$
4. d. $$ \frac{a+1}{a-6}, a \neq 3, 6 $$
5. d. $$ \frac{n-6}{(n+1)(n+8)} $$
1. c. $$ \frac{x-2}{x+5} $$; where $$ x \neq -5 $$ 2. c. $$ -\frac{x+8}{10x} $$; where $$ x \neq 0, 8 $$ 3. b. $$ \frac{4a}{7b^{2}}, a \neq 0, b \neq 0 $$ 4. d. $$ \frac{a+1}{a-6}, a \neq 3, 6 $$ 5. d. $$ \frac{n-6}{(n+1)(n+8)} $$

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