Simplify each complex fraction: A math worksheet with eight problems involving algebraic expressions and fractions.

To simplify each of the given complex fractions, we will follow a systematic approach. Complex fractions are fractions where the numerator, denominator, or both contain fractions. The general strategy is to simplify the numerator and denominator separately (if possible) and then divide them by multiplying by the reciprocal of the denominator.

Let's solve each problem step by step.

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1. Simplify:

$$
\frac{\frac{2x}{27y^2}}{\frac{6x^2}{9}}
$$

#### Step 1: Rewrite the division as multiplication by the reciprocal.
$$
\frac{\frac{2x}{27y^2}}{\frac{6x^2}{9}} = \frac{2x}{27y^2} \cdot \frac{9}{6x^2}
$$

#### Step 2: Simplify the expression.
$$
\frac{2x \cdot 9}{27y^2 \cdot 6x^2} = \frac{18x}{162x^2y^2}
$$

#### Step 3: Cancel common factors.
- The numerator has $18x$.
- The denominator has $162x^2y^2$.
- Factor out common terms: $18x$ divides both numerator and denominator.

$$
\frac{18x}{162x^2y^2} = \frac{1}{9xy^2}
$$

#### Final Answer:
$$
\boxed{\frac{1}{9xy^2}}
$$

---

2. Simplify:

$$
\frac{\frac{5x}{10}}{\frac{x+2}{x-2}}
$$

#### Step 1: Simplify the numerator.
$$
\frac{5x}{10} = \frac{x}{2}
$$

#### Step 2: Rewrite the division as multiplication by the reciprocal.
$$
\frac{\frac{x}{2}}{\frac{x+2}{x-2}} = \frac{x}{2} \cdot \frac{x-2}{x+2}
$$

#### Step 3: Multiply the fractions.
$$
\frac{x}{2} \cdot \frac{x-2}{x+2} = \frac{x(x-2)}{2(x+2)}
$$

#### Final Answer:
$$
\boxed{\frac{x(x-2)}{2(x+2)}}
$$

---

3. Simplify:

$$
\frac{x + \frac{1}{y}}{y + \frac{1}{x}}
$$

#### Step 1: Simplify the numerator.
$$
x + \frac{1}{y} = \frac{xy}{y} + \frac{1}{y} = \frac{xy + 1}{y}
$$

#### Step 2: Simplify the denominator.
$$
y + \frac{1}{x} = \frac{yx}{x} + \frac{1}{x} = \frac{yx + 1}{x}
$$

#### Step 3: Rewrite the division as multiplication by the reciprocal.
$$
\frac{x + \frac{1}{y}}{y + \frac{1}{x}} = \frac{\frac{xy + 1}{y}}{\frac{yx + 1}{x}} = \frac{xy + 1}{y} \cdot \frac{x}{yx + 1}
$$

#### Step 4: Multiply the fractions.
$$
\frac{xy + 1}{y} \cdot \frac{x}{yx + 1} = \frac{x(xy + 1)}{y(yx + 1)}
$$

#### Final Answer:
$$
\boxed{\frac{x(xy + 1)}{y(yx + 1)}}
$$

---

4. Simplify:

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

#### Step 1: Simplify the numerator.
$$
\frac{x^2 - 2}{x} = x - \frac{2}{x}
$$

#### Step 2: Simplify the denominator.
$$
\frac{1}{x} + 2 = \frac{1}{x} + \frac{2x}{x} = \frac{1 + 2x}{x}
$$

#### Step 3: Rewrite the division as multiplication by the reciprocal.
$$
\frac{\frac{x^2 - 2}{x}}{\frac{1}{x} + 2} = \frac{x^2 - 2}{x} \cdot \frac{x}{1 + 2x}
$$

#### Step 4: Multiply the fractions.
$$
\frac{x^2 - 2}{x} \cdot \frac{x}{1 + 2x} = \frac{(x^2 - 2)x}{x(1 + 2x)} = \frac{x^2 - 2}{1 + 2x}
$$

#### Final Answer:
$$
\boxed{\frac{x^2 - 2}{1 + 2x}}
$$

---

5. Simplify:

$$
\frac{\frac{x + 1}{2x - 1}}{6}
$$

#### Step 1: Rewrite the division as multiplication by the reciprocal.
$$
\frac{\frac{x + 1}{2x - 1}}{6} = \frac{x + 1}{2x - 1} \cdot \frac{1}{6}
$$

#### Step 2: Multiply the fractions.
$$
\frac{x + 1}{2x - 1} \cdot \frac{1}{6} = \frac{x + 1}{6(2x - 1)}
$$

#### Final Answer:
$$
\boxed{\frac{x + 1}{6(2x - 1)}}
$$

---

6. Simplify:

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

#### Step 1: Simplify the numerator.
$$
2 + \frac{1}{x} = \frac{2x}{x} + \frac{1}{x} = \frac{2x + 1}{x}
$$

#### Step 2: Simplify the denominator.
$$
4x - \frac{1}{x} = \frac{4x^2}{x} - \frac{1}{x} = \frac{4x^2 - 1}{x}
$$

#### Step 3: Rewrite the division as multiplication by the reciprocal.
$$
\frac{2 + \frac{1}{x}}{4x - \frac{1}{x}} = \frac{\frac{2x + 1}{x}}{\frac{4x^2 - 1}{x}} = \frac{2x + 1}{x} \cdot \frac{x}{4x^2 - 1}
$$

#### Step 4: Multiply the fractions.
$$
\frac{2x + 1}{x} \cdot \frac{x}{4x^2 - 1} = \frac{(2x + 1)x}{x(4x^2 - 1)} = \frac{2x + 1}{4x^2 - 1}
$$

#### Step 5: Factor the denominator if possible.
$$
4x^2 - 1 = (2x - 1)(2x + 1)
$$

#### Step 6: Simplify further.
$$
\frac{2x + 1}{(2x - 1)(2x + 1)} = \frac{1}{2x - 1}
$$

#### Final Answer:
$$
\boxed{\frac{1}{2x - 1}}
$$

---

7. Simplify:

$$
\frac{\frac{x + 3}{12}}{\frac{4x - 5}{15}}
$$

#### Step 1: Rewrite the division as multiplication by the reciprocal.
$$
\frac{\frac{x + 3}{12}}{\frac{4x - 5}{15}} = \frac{x + 3}{12} \cdot \frac{15}{4x - 5}
$$

#### Step 2: Multiply the fractions.
$$
\frac{x + 3}{12} \cdot \frac{15}{4x - 5} = \frac{(x + 3) \cdot 15}{12 \cdot (4x - 5)} = \frac{15(x + 3)}{12(4x - 5)}
$$

#### Step 3: Simplify the constants.
$$
\frac{15(x + 3)}{12(4x - 5)} = \frac{5(x + 3)}{4(4x - 5)}
$$

#### Final Answer:
$$
\boxed{\frac{5(x + 3)}{4(4x - 5)}}
$$

---

8. Simplify:

$$
\frac{\frac{5}{1} - \frac{1}{3}}{\frac{6}{1} + \frac{2}{8}}
$$

#### Step 1: Simplify the numerator.
$$
\frac{5}{1} - \frac{1}{3} = \frac{15}{3} - \frac{1}{3} = \frac{14}{3}
$$

#### Step 2: Simplify the denominator.
$$
\frac{6}{1} + \frac{2}{8} = \frac{6}{1} + \frac{1}{4} = \frac{24}{4} + \frac{1}{4} = \frac{25}{4}
$$

#### Step 3: Rewrite the division as multiplication by the reciprocal.
$$
\frac{\frac{14}{3}}{\frac{25}{4}} = \frac{14}{3} \cdot \frac{4}{25}
$$

#### Step 4: Multiply the fractions.
$$
\frac{14}{3} \cdot \frac{4}{25} = \frac{14 \cdot 4}{3 \cdot 25} = \frac{56}{75}
$$

#### Final Answer:
$$
\boxed{\frac{56}{75}}
$$

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Final Answers:

1. $\boxed{\frac{1}{9xy^2}}$
2. $\boxed{\frac{x(x-2)}{2(x+2)}}$
3. $\boxed{\frac{x(xy + 1)}{y(yx + 1)}}$
4. $\boxed{\frac{x^2 - 2}{1 + 2x}}$
5. $\boxed{\frac{x + 1}{6(2x - 1)}}$
6. $\boxed{\frac{1}{2x - 1}}$
7. $\boxed{\frac{5(x + 3)}{4(4x - 5)}}$
8. $\boxed{\frac{56}{75}}$