Students can use this worksheet to practice simplifying rational expressions involving multiplication and division.

(1) $\frac{1}{x - 1} \times \frac{8x - 8}{8}$

Factor the numerator: $8x - 8 = 8(x - 1)$

$\frac{1}{x - 1} \times \frac{8(x - 1)}{8}$

Cancel common factors: $(x - 1)$ and $8$

$\frac{1}{\cancel{x - 1}} \times \frac{\cancel{8}(x - 1)}{\cancel{8}} = 1$

Answer: $1$

(2) $\frac{2y - 5}{(y + 2)(y - 3)} \times \frac{y - 3}{4y - 10}$

Factor the denominator of the second fraction: $4y - 10 = 2(2y - 5)$

$\frac{2y - 5}{(y + 2)(y - 3)} \times \frac{y - 3}{2(2y - 5)}$

Cancel common factors: $(2y - 5)$ and $(y - 3)$

$\frac{\cancel{2y - 5}}{(y + 2)\cancel{(y - 3)}} \times \frac{\cancel{y - 3}}{2\cancel{(2y - 5)}} = \frac{1}{2(y + 2)}$

Answer: $\frac{1}{2(y + 2)}$

(3) $\frac{3x - 9y}{x^2 - xy} \div \frac{x^2 - 9y^2}{x^2 - y^2}$

First, rewrite division as multiplication by the reciprocal:

$\frac{3x - 9y}{x^2 - xy} \times \frac{x^2 - y^2}{x^2 - 9y^2}$

Factor each expression:

- $3x - 9y = 3(x - 3y)$
- $x^2 - xy = x(x - y)$
- $x^2 - y^2 = (x - y)(x + y)$
- $x^2 - 9y^2 = (x - 3y)(x + 3y)$

Substitute:

$\frac{3(x - 3y)}{x(x - y)} \times \frac{(x - y)(x + y)}{(x - 3y)(x + 3y)}$

Cancel common factors: $(x - 3y)$ and $(x - y)$

$\frac{3\cancel{(x - 3y)}}{x\cancel{(x - y)}} \times \frac{\cancel{(x - y)}(x + y)}{\cancel{(x - 3y)}(x + 3y)} = \frac{3(x + y)}{x(x + 3y)}$

Answer: $\frac{3(x + y)}{x(x + 3y)}$

(4) $\frac{r - 2}{8r + 16} \times \frac{r + 2}{r^2 - 2r}$

Factor each expression:

- $8r + 16 = 8(r + 2)$
- $r^2 - 2r = r(r - 2)$

Substitute:

$\frac{r - 2}{8(r + 2)} \times \frac{r + 2}{r(r - 2)}$

Cancel common factors: $(r - 2)$ and $(r + 2)$

$\frac{\cancel{r - 2}}{8\cancel{(r + 2)}} \times \frac{\cancel{r + 2}}{r\cancel{(r - 2)}} = \frac{1}{8r}$

Answer: $\frac{1}{8r}$

(5) $\frac{1}{p - 4} \div \frac{3p}{4p - 16}$

Rewrite division as multiplication by the reciprocal:

$\frac{1}{p - 4} \times \frac{4p - 16}{3p}$

Factor: $4p - 16 = 4(p - 4)$

$\frac{1}{p - 4} \times \frac{4(p - 4)}{3p}$

Cancel $(p - 4)$

$\frac{1}{\cancel{p - 4}} \times \frac{4\cancel{(p - 4)}}{3p} = \frac{4}{3p}$

Answer: $\frac{4}{3p}$

(6) $\frac{x^3 + 2x^2}{y^3 - y} \times \frac{y^2 - 1}{x^2 - 4}$

Factor each expression:

- $x^3 + 2x^2 = x^2(x + 2)$
- $y^3 - y = y(y^2 - 1) = y(y - 1)(y + 1)$
- $y^2 - 1 = (y - 1)(y + 1)$
- $x^2 - 4 = (x - 2)(x + 2)$

Substitute:

$\frac{x^2(x + 2)}{y(y - 1)(y + 1)} \times \frac{(y - 1)(y + 1)}{(x - 2)(x + 2)}$

Cancel common factors: $(x + 2)$, $(y - 1)$, $(y + 1)$

$\frac{x^2\cancel{(x + 2)}}{y\cancel{(y - 1)}\cancel{(y + 1)}} \times \frac{\cancel{(y - 1)}\cancel{(y + 1)}}{(x - 2)\cancel{(x + 2)}} = \frac{x^2}{y(x - 2)}$

Answer: $\frac{x^2}{y(x - 2)}$

(7) $\frac{5b^2c^2}{10bc} \div \frac{6b^2c}{2c^2}$

Rewrite division as multiplication by the reciprocal:

$\frac{5b^2c^2}{10bc} \times \frac{2c^2}{6b^2c}$

Simplify each fraction:

- $\frac{5b^2c^2}{10bc} = \frac{b c}{2}$
- $\frac{2c^2}{6b^2c} = \frac{c}{3b^2}$

Multiply:

$\frac{b c}{2} \times \frac{c}{3b^2} = \frac{b c \cdot c}{2 \cdot 3b^2} = \frac{b c^2}{6b^2} = \frac{c^2}{6b}$

Answer: $\frac{c^2}{6b}$

(8) $\frac{2p}{q} \div \frac{4p}{q^2}$

Rewrite division as multiplication by the reciprocal:

$\frac{2p}{q} \times \frac{q^2}{4p}$

Cancel common factors: $2p$ and $q$

$\frac{2p}{q} \times \frac{q^2}{4p} = \frac{2p \cdot q^2}{q \cdot 4p} = \frac{2q}{4} = \frac{q}{2}$

Answer: $\frac{q}{2}$