Math worksheet for simplifying rational expressions, featuring multiplication and division problems.

Here are the step-by-step solutions for each problem on the worksheet.

1) Simplify: $\frac{5q + 5}{q^2 + 19q + 88} \cdot \frac{q + 8}{5q + 5}$

* Step 1: Factor the numerators and denominators.
* $5q + 5$ factors to $5(q + 1)$.
* $q^2 + 19q + 88$ factors to $(q + 11)(q + 8)$ because $11 \cdot 8 = 88$ and $11 + 8 = 19$.
* Step 2: Rewrite the expression:
$$ \frac{5(q + 1)}{(q + 11)(q + 8)} \cdot \frac{q + 8}{5(q + 1)} $$
* Step 3: Cancel common terms from the top and bottom.
* The $5(q+1)$ on the top left cancels with the $5(q+1)$ on the bottom right.
* The $(q+8)$ on the top right cancels with the $(q+8)$ on the bottom left.
* Step 4: What is left?
Only $1$ on top and $(q + 11)$ on the bottom.

2) Simplify: $\frac{9}{11} \cdot \frac{2}{5n}$

* Step 1: Multiply straight across.
* Numerator: $9 \cdot 2 = 18$
* Denominator: $11 \cdot 5n = 55n$
* Step 2: Check if it simplifies. 18 and 55 share no common factors.

3) Simplify: $\frac{y^2 + 20y + 96}{y^2 + 14y + 24} \div \frac{1}{y + 2}$

* Step 1: Change division to multiplication by flipping the second fraction (reciprocal).
$$ \frac{y^2 + 20y + 96}{y^2 + 14y + 24} \cdot \frac{y + 2}{1} $$
* Step 2: Factor the quadratic expressions.
* $y^2 + 20y + 96$: Find numbers that multiply to 96 and add to 20. Those are 12 and 8. So, $(y + 12)(y + 8)$.
* $y^2 + 14y + 24$: Find numbers that multiply to 24 and add to 14. Those are 12 and 2. So, $(y + 12)(y + 2)$.
* Step 3: Rewrite and cancel.
$$ \frac{(y + 12)(y + 8)}{(y + 12)(y + 2)} \cdot \frac{y + 2}{1} $$
* Cancel $(y + 12)$ from top and bottom.
* Cancel $(y + 2)$ from top and bottom.
* Step 4: What is left? Just $(y + 8)$.

4) Simplify: $\frac{30g^2 - 25g - 20}{33g^2 - 20g - 32} \div \frac{g^2}{66g^2 - 84g - 96}$

* Step 1: Flip the second fraction to multiply.
$$ \frac{30g^2 - 25g - 20}{33g^2 - 20g - 32} \cdot \frac{66g^2 - 84g - 96}{g^2} $$
* Step 2: Factor everything completely.
* Top Left: $30g^2 - 25g - 20$. Factor out 5 first: $5(6g^2 - 5g - 4)$. Then factor the inside: $5(3g - 4)(2g + 1)$.
* Bottom Left: $33g^2 - 20g - 32$. Factors to $(3g - 4)(11g + 8)$.
* Top Right: $66g^2 - 84g - 96$. Factor out 6 first: $6(11g^2 - 14g - 16)$. Then factor the inside: $6(11g + 8)(g - 2)$.
* Bottom Right: $g^2$ stays as is.
* Step 3: Combine and cancel.
$$ \frac{5(3g - 4)(2g + 1) \cdot 6(11g + 8)(g - 2)}{(3g - 4)(11g + 8) \cdot g^2} $$
* Cancel $(3g - 4)$.
* Cancel $(11g + 8)$.
* Multiply remaining numbers: $5 \cdot 6 = 30$.
* Step 4: Final result: $\frac{30(2g + 1)(g - 2)}{g^2}$

5) Simplify: $\frac{s^2 + 3s + 2}{7s} \div (s + 1)$

* Step 1: Treat $(s+1)$ as a fraction $\frac{s+1}{1}$ and flip it.
$$ \frac{s^2 + 3s + 2}{7s} \cdot \frac{1}{s + 1} $$
* Step 2: Factor the numerator $s^2 + 3s + 2$. It becomes $(s + 2)(s + 1)$.
* Step 3: Cancel terms.
$$ \frac{(s + 2)(s + 1)}{7s} \cdot \frac{1}{s + 1} $$
* The $(s + 1)$ on top cancels with the $(s + 1)$ on bottom.
* Step 4: Final result: $\frac{s + 2}{7s}$

6) Simplify: $\frac{6d^2}{4} \div \frac{11d}{2}$

* Step 1: Flip and multiply.
$$ \frac{6d^2}{4} \cdot \frac{2}{11d} $$
* Step 2: Simplify before multiplying.
* $\frac{6d^2}{4}$ simplifies to $\frac{3d^2}{2}$.
* Now we have: $\frac{3d^2}{2} \cdot \frac{2}{11d}$
* Step 3: Cancel the 2s and one $d$.
* The 2 in the denominator cancels the 2 in the numerator.
* $d^2$ divided by $d$ leaves just $d$ on top.
* Step 4: Final result: $\frac{3d}{11}$

7) Simplify: $\frac{8(r - 4)}{(r - 4)} \cdot \frac{9r}{8(r - 2)}$

* Step 1: Identify common terms to cancel immediately.
* The $(r - 4)$ on top and bottom of the first fraction cancel out.
* The $8$ on top of the first fraction and the $8$ on the bottom of the second fraction cancel out.
* Step 2: Look at what remains.
* Top: $1 \cdot 9r = 9r$
* Bottom: $1 \cdot (r - 2) = r - 2$
* Step 3: Final result: $\frac{9r}{r - 2}$

8) Simplify: $\frac{10}{(h + 4)} \cdot \frac{11h - 77}{(h - 7)}$

* Step 1: Factor the numerator $11h - 77$.
* Factor out 11: $11(h - 7)$.
* Step 2: Rewrite the expression.
$$ \frac{10}{h + 4} \cdot \frac{11(h - 7)}{h - 7} $$
* Step 3: Cancel the $(h - 7)$ terms.
* Step 4: Multiply what is left.
* Top: $10 \cdot 11 = 110$
* Bottom: $h + 4$
* Step 5: Final result: $\frac{110}{h + 4}$

9) Simplify: $(z - 3) \cdot \frac{7z}{z^2 - 4z + 3}$

* Step 1: Write $(z-3)$ as a fraction: $\frac{z-3}{1}$.
* Step 2: Factor the denominator $z^2 - 4z + 3$.
* Numbers that multiply to 3 and add to -4 are -3 and -1.
* So, $(z - 3)(z - 1)$.
* Step 3: Set up the multiplication.
$$ \frac{z - 3}{1} \cdot \frac{7z}{(z - 3)(z - 1)} $$
* Step 4: Cancel the $(z - 3)$ terms.
* Step 5: Final result: $\frac{7z}{z - 1}$

10) Simplify: $\frac{4(b + 10)}{10} \cdot \frac{2b}{4(b + 10)}$

* Step 1: Look for identical terms to cancel.
* The term $4(b + 10)$ appears in the top left and the bottom right. They cancel each other out completely (become 1).
* Step 2: Look at what remains.
* Top: $2b$
* Bottom: $10$
* Step 3: Simplify the fraction $\frac{2b}{10}$.
* Divide top and bottom by 2.
* Step 4: Final result: $\frac{b}{5}$

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Final Answer:

1) $\frac{1}{q + 11}$
2) $\frac{18}{55n}$
3) $y + 8$
4) $\frac{30(2g + 1)(g - 2)}{g^2}$
5) $\frac{s + 2}{7s}$
6) $\frac{3d}{11}$
7) $\frac{9r}{r - 2}$
8) $\frac{110}{h + 4}$
9) $\frac{7z}{z - 1}$
10) $\frac{b}{5}$