SOLUTION: Multiplying and Dividing Rational Expressions Worksheet ... - Free Printable

Problem: Simplify each expression (Multiplying and Dividing Rational Expressions)

The worksheet contains several problems involving the multiplication and division of rational expressions. Below, I will solve a few representative problems from the list to demonstrate the process. If you need solutions for specific problems, please let me know!

---

#### Problem 1:
Simplify:
\[
\frac{12x}{14} \times \frac{14}{16x}
\]

Solution:
1. Write the expression:
\[
\frac{12x}{14} \times \frac{14}{16x}
\]

2. Multiply the numerators and denominators:
\[
\frac{12x \cdot 14}{14 \cdot 16x}
\]

3. Simplify by canceling common factors:
- The \(14\) in the numerator and denominator cancel out.
- The \(x\) in the numerator and denominator cancel out.
- Simplify \(12 / 16\) to \(3 / 4\):
\[
\frac{12x \cdot 14}{14 \cdot 16x} = \frac{12}{16} = \frac{3}{4}
\]

4. Final answer:
\[
\boxed{\frac{3}{4}}
\]

---

#### Problem 9:
Simplify:
\[
\frac{1}{x+10} \times \frac{10x+30}{x+3}
\]

Solution:
1. Write the expression:
\[
\frac{1}{x+10} \times \frac{10x+30}{x+3}
\]

2. Factor the numerator \(10x + 30\):
\[
10x + 30 = 10(x + 3)
\]
So the expression becomes:
\[
\frac{1}{x+10} \times \frac{10(x+3)}{x+3}
\]

3. Cancel the common factor \((x+3)\) in the numerator and denominator:
\[
\frac{1}{x+10} \times \frac{10(x+3)}{x+3} = \frac{1 \cdot 10}{x+10} = \frac{10}{x+10}
\]

4. Final answer:
\[
\boxed{\frac{10}{x+10}}
\]

---

#### Problem 15:
Simplify:
\[
\frac{1}{x-9} \times \frac{x^2 + 6x - 27}{x+9}
\]

Solution:
1. Write the expression:
\[
\frac{1}{x-9} \times \frac{x^2 + 6x - 27}{x+9}
\]

2. Factor the quadratic expression \(x^2 + 6x - 27\):
\[
x^2 + 6x - 27 = (x + 9)(x - 3)
\]
So the expression becomes:
\[
\frac{1}{x-9} \times \frac{(x+9)(x-3)}{x+9}
\]

3. Cancel the common factor \((x+9)\) in the numerator and denominator:
\[
\frac{1}{x-9} \times \frac{(x+9)(x-3)}{x+9} = \frac{1 \cdot (x-3)}{x-9} = \frac{x-3}{x-9}
\]

4. Final answer:
\[
\boxed{\frac{x-3}{x-9}}
\]

---

#### Problem 16:
Simplify:
\[
\frac{x^2 - 10x + 25}{10x - 100} \times \frac{x-10}{45-9x}
\]

Solution:
1. Write the expression:
\[
\frac{x^2 - 10x + 25}{10x - 100} \times \frac{x-10}{45-9x}
\]

2. Factor each part:
- Factor \(x^2 - 10x + 25\):
\[
x^2 - 10x + 25 = (x-5)^2
\]
- Factor \(10x - 100\):
\[
10x - 100 = 10(x-10)
\]
- Factor \(45 - 9x\):
\[
45 - 9x = -9(x-5)
\]
So the expression becomes:
\[
\frac{(x-5)^2}{10(x-10)} \times \frac{x-10}{-9(x-5)}
\]

3. Simplify by canceling common factors:
- Cancel one \((x-5)\) from the numerator and denominator.
- Cancel \((x-10)\) from the numerator and denominator.
\[
\frac{(x-5)^2}{10(x-10)} \times \frac{x-10}{-9(x-5)} = \frac{x-5}{10} \times \frac{1}{-9} = \frac{x-5}{-90} = -\frac{x-5}{90}
\]

4. Final answer:
\[
\boxed{-\frac{x-5}{90}}
\]

---

Summary of Solutions:

- Problem 1: \(\boxed{\frac{3}{4}}\)
- Problem 9: \(\boxed{\frac{10}{x+10}}\)
- Problem 15: \(\boxed{\frac{x-3}{x-9}}\)
- Problem 16: \(\boxed{-\frac{x-5}{90}}\)

If you need solutions for other problems, please specify!