Let's go through the worksheet step by step, starting with question 1 and simplifying each expression.

1) Simplify the following:

a) $\frac{5x}{2y^2} + \frac{1}{6y^3}$

• To add these two terms, we need to find a common denominator. The least common denominator (LCD) is $6y^3$.
• The first term becomes: $\frac{5x}{2y^2} = \frac{15x}{6y^3}$.
• Now, we have $\frac{15x}{6y^3} + \frac{1}{6y^3} = \frac{15x + 1}{6y^3}$.

b) $\frac{1}{4k^3} + \frac{5}{8y^3k^2} - \frac{3}{20yk^2}$

• The least common denominator is $40k^3y^3$.
• Adjust each fraction to have this denominator and combine them.

c) $\frac{3}{2x - 1} + \frac{4x}{x + 5}$

• Find the least common denominator: $(2x - 1)(x + 5)$.
• Rewrite both fractions with this denominator and simplify the result.

d) $\frac{5}{x^2 - 3x} + \frac{4x + 1}{x^2 - 9}$

• Factor both denominators:
  • $x^2 - 3x = x(x - 3)$,
  • $x^2 - 9 = (x - 3)(x + 3)$.
• Use the least common denominator: $x(x - 3)(x + 3)$, and combine the two fractions.

e) $\frac{1}{4x + 6} + \frac{x + 2}{4x^2 - 9}$

• Factor the denominators:
  • $4x + 6 = 2(2x + 3)$,
  • $4x^2 - 9 = (2x + 3)(2x - 3)$.
• Combine the two fractions with the least common denominator.

f) $\frac{2x + 1}{x^2 - 3x - 10} + \frac{4}{x^2 - 25}$

• Factor the denominators:
  • $x^2 - 3x - 10 = (x - 5)(x + 2)$,
  • $x^2 - 25 = (x - 5)(x + 5)$.
• Use the least common denominator: $(x - 5)(x + 2)(x + 5)$, and combine the fractions.

g) $\frac{2x + 1}{5 - 4x - x^2} + \frac{3}{x^3 + 5x^2}$

• Factor both denominators. For the first, rewrite it as $-(x + 5)(x + 1)$.
• The second can be factored as $x^2(x + 5)$.
• Find a common denominator and combine the fractions.

h) $\frac{3}{x^2 - 4} + \frac{x + 5}{x^3 + 8}$

• Factor the denominators:
  • $x^2 - 4 = (x - 2)(x + 2)$,
  • $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$.
• Combine the fractions with the least common denominator.

2) Simplify the following complex fraction:

$\frac{\frac{1}{k^2 - 4} + \frac{3}{k + 2}}{\frac{1}{k + 2} + \frac{5}{k - 2}}$

• First, simplify the numerator:
  • $\frac{1}{k^2 - 4} = \frac{1}{(k - 2)(k + 2)}$.
  • Combine the fractions in the numerator with the least common denominator $(k - 2)(k + 2)$.
• Similarly, simplify the denominator by finding a common denominator for the fractions.

Would you like detailed steps for any of these problems, or should I proceed with solving any particular one in more detail?

Here are 5 related questions to consider:

1. What is the process for finding the least common denominator between two fractions?
2. How can factoring help simplify complex rational expressions?
3. What are some common factoring techniques useful in simplifying algebraic fractions?
4. How do you deal with complex fractions where both the numerator and the denominator are fractions themselves?
5. What is the role of common factors when simplifying a fraction?

Tip: Always check for common factors before attempting to find the least common denominator; it often makes the process easier!