Algebra worksheet for simplifying complex rational expressions, featuring 12 problems with fractions and variables.

To simplify the given complex rational expressions, we will follow a systematic approach. Here are the solutions for each problem:

---

Problem 1:

$$
\frac{\frac{u+2}{4} - \frac{2}{u+2}}{u+2}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{u+2}{4} - \frac{2}{u+2}
$$
Find a common denominator, which is \(4(u+2)\):
$$
\frac{(u+2)^2}{4(u+2)} - \frac{8}{4(u+2)} = \frac{(u+2)^2 - 8}{4(u+2)}
$$
Expand \((u+2)^2\):
$$
(u+2)^2 = u^2 + 4u + 4
$$
So the numerator becomes:
$$
\frac{u^2 + 4u + 4 - 8}{4(u+2)} = \frac{u^2 + 4u - 4}{4(u+2)}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{u^2 + 4u - 4}{4(u+2)}}{u+2} = \frac{u^2 + 4u - 4}{4(u+2)^2}
$$

#### Final Answer:
$$
\boxed{\frac{u^2 + 4u - 4}{4(u+2)^2}}
$$

---

Problem 2:

$$
\frac{\frac{x}{2} - \frac{2}{x^2}}{\frac{x-4}{x^2}}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{x}{2} - \frac{2}{x^2}
$$
Find a common denominator, which is \(2x^2\):
$$
\frac{x^3}{2x^2} - \frac{4}{2x^2} = \frac{x^3 - 4}{2x^2}
$$

#### Step 2: Simplify the denominator.
The denominator is:
$$
\frac{x-4}{x^2}
$$

#### Step 3: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{x^3 - 4}{2x^2}}{\frac{x-4}{x^2}} = \frac{x^3 - 4}{2x^2} \cdot \frac{x^2}{x-4} = \frac{x^3 - 4}{2(x-4)}
$$

#### Final Answer:
$$
\boxed{\frac{x^3 - 4}{2(x-4)}}
$$

---

Problem 3:

$$
\frac{\frac{4}{x}}{\frac{1}{2} - \frac{x^2}{2}}
$$

#### Step 1: Simplify the denominator.
The denominator is:
$$
\frac{1}{2} - \frac{x^2}{2}
$$
Find a common denominator, which is \(2\):
$$
\frac{1 - x^2}{2}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{4}{x}}{\frac{1 - x^2}{2}} = \frac{4}{x} \cdot \frac{2}{1 - x^2} = \frac{8}{x(1 - x^2)}
$$

#### Final Answer:
$$
\boxed{\frac{8}{x(1 - x^2)}}
$$

---

Problem 4:

$$
\frac{\frac{2}{x-3} + \frac{4}{x-3}}{\frac{2}{x^2}}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{2}{x-3} + \frac{4}{x-3}
$$
Combine the fractions:
$$
\frac{2 + 4}{x-3} = \frac{6}{x-3}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{6}{x-3}}{\frac{2}{x^2}} = \frac{6}{x-3} \cdot \frac{x^2}{2} = \frac{6x^2}{2(x-3)} = \frac{3x^2}{x-3}
$$

#### Final Answer:
$$
\boxed{\frac{3x^2}{x-3}}
$$

---

Problem 5:

$$
\frac{\frac{16}{x^2} + \frac{2}{x^2}}{x^2}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{16}{x^2} + \frac{2}{x^2}
$$
Combine the fractions:
$$
\frac{16 + 2}{x^2} = \frac{18}{x^2}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{18}{x^2}}{x^2} = \frac{18}{x^2} \cdot \frac{1}{x^2} = \frac{18}{x^4}
$$

#### Final Answer:
$$
\boxed{\frac{18}{x^4}}
$$

---

Problem 6:

$$
\frac{\frac{4m}{16} - \frac{16}{m}}{3}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{4m}{16} - \frac{16}{m}
$$
Simplify \(\frac{4m}{16}\):
$$
\frac{4m}{16} = \frac{m}{4}
$$
So the numerator becomes:
$$
\frac{m}{4} - \frac{16}{m}
$$
Find a common denominator, which is \(4m\):
$$
\frac{m^2}{4m} - \frac{64}{4m} = \frac{m^2 - 64}{4m}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{m^2 - 64}{4m}}{3} = \frac{m^2 - 64}{4m} \cdot \frac{1}{3} = \frac{m^2 - 64}{12m}
$$

#### Final Answer:
$$
\boxed{\frac{m^2 - 64}{12m}}
$$

---

Problem 7:

$$
\frac{\frac{x^2}{4} - \frac{x}{4}}{x}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{x^2}{4} - \frac{x}{4}
$$
Combine the fractions:
$$
\frac{x^2 - x}{4}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{x^2 - x}{4}}{x} = \frac{x^2 - x}{4} \cdot \frac{1}{x} = \frac{x(x-1)}{4} \cdot \frac{1}{x} = \frac{x-1}{4}
$$

#### Final Answer:
$$
\boxed{\frac{x-1}{4}}
$$

---

Problem 8:

$$
\frac{\frac{4}{x^2} - \frac{3}{x}}{\frac{9}{x}}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{4}{x^2} - \frac{3}{x}
$$
Find a common denominator, which is \(x^2\):
$$
\frac{4}{x^2} - \frac{3x}{x^2} = \frac{4 - 3x}{x^2}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{4 - 3x}{x^2}}{\frac{9}{x}} = \frac{4 - 3x}{x^2} \cdot \frac{x}{9} = \frac{4 - 3x}{9x}
$$

#### Final Answer:
$$
\boxed{\frac{4 - 3x}{9x}}
$$

---

Problem 9:

$$
\frac{\frac{1}{x} + \frac{1}{2}}{4}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{1}{x} + \frac{1}{2}
$$
Find a common denominator, which is \(2x\):
$$
\frac{2}{2x} + \frac{x}{2x} = \frac{2 + x}{2x}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{2 + x}{2x}}{4} = \frac{2 + x}{2x} \cdot \frac{1}{4} = \frac{2 + x}{8x}
$$

#### Final Answer:
$$
\boxed{\frac{2 + x}{8x}}
$$

---

Problem 10:

$$
\frac{\frac{5}{m-1} + \frac{5}{m-1}}{m-1}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{5}{m-1} + \frac{5}{m-1}
$$
Combine the fractions:
$$
\frac{5 + 5}{m-1} = \frac{10}{m-1}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{10}{m-1}}{m-1} = \frac{10}{m-1} \cdot \frac{1}{m-1} = \frac{10}{(m-1)^2}
$$

#### Final Answer:
$$
\boxed{\frac{10}{(m-1)^2}}
$$

---

Problem 11:

$$
\frac{\frac{x-4}{36} - \frac{6}{x-4}}{x-4}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{x-4}{36} - \frac{6}{x-4}
$$
Find a common denominator, which is \(36(x-4)\):
$$
\frac{(x-4)^2}{36(x-4)} - \frac{216}{36(x-4)} = \frac{(x-4)^2 - 216}{36(x-4)}
$$
Expand \((x-4)^2\):
$$
(x-4)^2 = x^2 - 8x + 16
$$
So the numerator becomes:
$$
\frac{x^2 - 8x + 16 - 216}{36(x-4)} = \frac{x^2 - 8x - 200}{36(x-4)}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{x^2 - 8x - 200}{36(x-4)}}{x-4} = \frac{x^2 - 8x - 200}{36(x-4)^2}
$$

#### Final Answer:
$$
\boxed{\frac{x^2 - 8x - 200}{36(x-4)^2}}
$$

---

Problem 12:

$$
\frac{\frac{5x}{36} - \frac{5}{x}}{\frac{x^2}{6}}
$$

#### Step 1: Simplify the numerator.
The numerator is:
$$
\frac{5x}{36} - \frac{5}{x}
$$
Find a common denominator, which is \(36x\):
$$
\frac{5x^2}{36x} - \frac{180}{36x} = \frac{5x^2 - 180}{36x}
$$

#### Step 2: Simplify the entire expression.
Now the expression is:
$$
\frac{\frac{5x^2 - 180}{36x}}{\frac{x^2}{6}} = \frac{5x^2 - 180}{36x} \cdot \frac{6}{x^2} = \frac{5(x^2 - 36)}{36x} \cdot \frac{6}{x^2} = \frac{5(x^2 - 36) \cdot 6}{36x \cdot x^2} = \frac{5(x^2 - 36)}{6x^3}
$$

#### Final Answer:
$$
\boxed{\frac{5(x^2 - 36)}{6x^3}}
$$

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Final Answers:

1. \(\boxed{\frac{u^2 + 4u - 4}{4(u+2)^2}}\)
2. \(\boxed{\frac{x^3 - 4}{2(x-4)}}\)
3. \(\boxed{\frac{8}{x(1 - x^2)}}\)
4. \(\boxed{\frac{3x^2}{x-3}}\)
5. \(\boxed{\frac{18}{x^4}}\)
6. \(\boxed{\frac{m^2 - 64}{12m}}\)
7. \(\boxed{\frac{x-1}{4}}\)
8. \(\boxed{\frac{4 - 3x}{9x}}\)
9. \(\boxed{\frac{2 + x}{8x}}\)
10. \(\boxed{\frac{10}{(m-1)^2}}\)
11. \(\boxed{\frac{x^2 - 8x - 200}{36(x-4)^2}}\)
12. \(\boxed{\frac{5(x^2 - 36)}{6x^3}}\)