Here are the simplified solutions for the rational expressions provided:

1. Simplify $\frac{129}{25} \cdot \frac{1}{50}$

$\frac{129}{25} \cdot \frac{1}{50} = \frac{129}{1250}$

This fraction is already in its simplest form as the numerator and denominator have no common factors.

Final Answer: $\frac{129}{1250}$

2. Simplify $\frac{3y + 9}{2} \cdot \frac{4}{6 - 3y}$

First, factorize $3y + 9$ and $6 - 3y$:

$3y + 9 = 3(y + 3), \quad 6 - 3y = 3(2 - y)$

The expression becomes:

$\frac{3(y + 3)}{2} \cdot \frac{4}{3(2 - y)}$

Simplify by canceling common terms:

$\frac{(y + 3)}{2} \cdot \frac{4}{2 - y}$

Note: $2 - y$ is equivalent to $-(y - 2)$. Adjust the sign:

$\frac{(y + 3)}{2} \cdot \frac{4}{-(y - 2)}$

$= \frac{4(y + 3)}{-2(y - 2)}$

Simplify:

$= \frac{-2(y + 3)}{(y - 2)}$

Final Answer: $\frac{-2(y + 3)}{y - 2}$

3. Simplify $\frac{z^2 - 32 - 10}{z^2 + 5z + 6} \cdot \frac{z^2 - 4}{z^2 - 8z + 15}$

Step 1: Factorize each part:

• $z^2 - 32 - 10 = z^2 - 42$ (Does not factor further.)
• $z^2 + 5z + 6 = (z + 2)(z + 3)$
• $z^2 - 4 = (z - 2)(z + 2)$ (Difference of squares)
• $z^2 - 8z + 15 = (z - 3)(z - 5)$

The expression becomes:

$\frac{z^2 - 42}{(z + 2)(z + 3)} \cdot \frac{(z - 2)(z + 2)}{(z - 3)(z - 5)}$

Step 2: Cancel common terms:

$= \frac{(z^2 - 42)(z - 2)}{(z + 3)(z - 5)}$

Final Answer: $\frac{(z^2 - 42)(z - 2)}{(z + 3)(z - 5)}$

4. Simplify $\frac{b^3 + 27}{6b + 30} \cdot \frac{3b^2 - 9b + 27}{b^3 + 80b + 15}$

Step 1: Factorize each part:

• $b^3 + 27 = (b + 3)(b^2 - 3b + 9)$ (Sum of cubes)
• $6b + 30 = 6(b + 5)$
• $3b^2 - 9b + 27 = 3(b^2 - 3b + 9)$
• $b^3 + 80b + 15$ (Does not factor easily.)

The expression becomes:

$\frac{(b + 3)(b^2 - 3b + 9)}{6(b + 5)} \cdot \frac{3(b^2 - 3b + 9)}{b^3 + 80b + 15}$

Step 2: Cancel common terms:

• $b^2 - 3b + 9$ appears in both numerator and denominator, so cancel it.

$= \frac{(b + 3) \cdot 3}{6(b + 5)(b^3 + 80b + 15)}$

Simplify the constant terms:

$= \frac{(b + 3)}{2(b + 5)(b^3 + 80b + 15)}$

Final Answer: $\frac{b + 3}{2(b + 5)(b^3 + 80b + 15)}$

Let me know if you have questions about any step or want further clarification!

Expanding your understanding:

1. How do you recognize and factor sum or difference of cubes?
2. Why is it important to identify common terms when simplifying rational expressions?
3. Can $b^3 + 80b + 15$ be factorized further? How would you approach it?
4. What are the restrictions on the variables in these expressions (denominator restrictions)?
5. How can factoring help simplify complex rational expressions quickly?

Tip: Always check for restrictions on the domain by identifying when the denominator equals zero.