Multiplying and dividing rational expressions worksheet with eight problems for practice.

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

Multiply

Problem 1:
$$ \frac{6x - 12}{x^2 - 9x + 18} \cdot \frac{7x - 21}{5x - 10} $$

1. Factor everything:
* Top left: $6(x - 2)$
* Bottom left: $(x - 3)(x - 6)$ (since $-3 \times -6 = 18$ and $-3 + -6 = -9$)
* Top right: $7(x - 3)$
* Bottom right: $5(x - 2)$
2. Rewrite the expression:
$$ \frac{6(x - 2)}{(x - 3)(x - 6)} \cdot \frac{7(x - 3)}{5(x - 2)} $$
3. Cancel common terms:
* $(x - 2)$ on top and bottom cancel out.
* $(x - 3)$ on top and bottom cancel out.
4. Multiply remaining parts:
* Numerator: $6 \cdot 7 = 42$
* Denominator: $5(x - 6)$

Answer: $\frac{42}{5(x - 6)}$ or $\frac{42}{5x - 30}$

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Problem 2:
$$ \frac{x^2 - 16}{x^2} \div \frac{x^2 - 4x}{x^2 + x - 12} $$

1. Flip the second fraction to change division into multiplication:
$$ \frac{x^2 - 16}{x^2} \cdot \frac{x^2 + x - 12}{x^2 - 4x} $$
2. Factor everything:
* Top left: $(x - 4)(x + 4)$ (difference of squares)
* Top right: $(x + 4)(x - 3)$ (factors of -12 that add to 1)
* Bottom right: $x(x - 4)$ (factor out x)
3. Rewrite and cancel:
$$ \frac{(x - 4)(x + 4)}{x \cdot x} \cdot \frac{(x + 4)(x - 3)}{x(x - 4)} $$
* Cancel $(x - 4)$ from top left and bottom right.
4. Combine remaining terms:
* Numerator: $(x + 4)(x + 4)(x - 3) = (x + 4)^2(x - 3)$
* Denominator: $x \cdot x \cdot x = x^3$

Answer: $\frac{(x + 4)^2(x - 3)}{x^3}$

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Problem 3:
$$ \frac{5t}{4t^2 - 8} \div \frac{6t - 12}{10t} $$

1. Flip to multiply:
$$ \frac{5t}{4t^2 - 8} \cdot \frac{10t}{6t - 12} $$
2. Factor denominators:
* Bottom left: $4(t^2 - 2)$
* Bottom right: $6(t - 2)$
*(Note: The first denominator $4t^2 - 8$ does not factor nicely with integers like a difference of squares unless we use roots, but looking at the pattern of other problems, it might be a typo for $4t^2 - 16$. However, solving exactly as written:)*

Let's re-examine Problem 3. Usually, these worksheets have clean integer factors.
If it is $4t^2 - 8$, we can't cancel easily with $6t-12$.
Let's assume standard factoring rules apply.
Numerator: $5t \cdot 10t = 50t^2$
Denominator: $(4t^2 - 8)(6t - 12) = 4(t^2 - 2) \cdot 6(t - 2) = 24(t^2 - 2)(t - 2)$

*Self-Correction*: Is it possible the first term was meant to be $4t^2 - 16$? If so, it would be $4(t-4)(t+4)$. No obvious cancellation.
What if the first term is $4t^2 - 8t$? Then $4t(t-2)$. That cancels perfectly! Given the context of "rational expressions", typos in user uploads or printouts happen. But I must solve what is there.

Let's look closer at the image. It says $4t^2 - 8$.
Okay, let's just simplify coefficients.
$$ \frac{50t^2}{(4t^2 - 8)(6t - 12)} $$
Factor out constants:
$$ \frac{50t^2}{4(t^2 - 2) \cdot 6(t - 2)} = \frac{50t^2}{24(t^2 - 2)(t - 2)} $$
Divide 50 and 24 by 2:
$$ \frac{25t^2}{12(t^2 - 2)(t - 2)} $$

*Alternative Interpretation*: Often in these specific worksheets, $4t^2 - 8$ is actually a typo for $4t^2 - 16$ which doesn't help, OR $4t^2 - 8t$.
However, looking at the visual spacing, it is definitely $4t^2 - 8$.
Wait, looking really closely at crop 3... is that a minus sign? Yes.
Let's stick to the literal math.
Result: $\frac{25t^2}{12(t^2 - 2)(t - 2)}$

*Wait, let me look at the problem again.*
Maybe the first denominator is $4t^2 - 8$ and the second numerator is $6t - 12$.
There is no common factor between $(t^2 - 2)$ and $(t - 2)$.

Let's try one more check. Is it possible the problem is:
$\frac{5t}{4t^2 - 8} \dots$
Actually, looking at similar "Brighterly" worksheets online, this specific problem often appears as:
$\frac{5t}{4t^2 - 8t} \div \frac{6t - 12}{10t}$
If it were $4t^2 - 8t$, then $4t(t-2)$. Then $6t-12 = 6(t-2)$.
Then $(t-2)$ cancels.
Since I cannot edit the image, I will provide the answer based on exactly what is written, but note that if it was a typo for $4t^2 - 8t$, the answer would be cleaner.

Strict Solution:
$$ \frac{5t \cdot 10t}{(4t^2 - 8)(6t - 12)} = \frac{50t^2}{24(t^2 - 2)(t - 2)} = \frac{25t^2}{12(t^2 - 2)(t - 2)} $$

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Problem 4:
$$ \frac{m^2 - n^2}{4m + 4n} \div \frac{m + n}{m \cdot n} $$

1. Flip to multiply:
$$ \frac{m^2 - n^2}{4m + 4n} \cdot \frac{mn}{m + n} $$
2. Factor:
* Top left: $(m - n)(m + n)$
* Bottom left: $4(m + n)$
3. Cancel:
* $(m + n)$ on top left and bottom left cancel.
* $(m + n)$ on top right cancels with nothing... wait.
Let's rewrite:
$$ \frac{(m - n)(m + n)}{4(m + n)} \cdot \frac{mn}{m + n} $$
Cancel one $(m+n)$ from the first fraction.
$$ \frac{m - n}{4} \cdot \frac{mn}{m + n} $$
Multiply straight across.
Numerator: $(m - n)mn$
Denominator: $4(m + n)$

Answer: $\frac{mn(m - n)}{4(m + n)}$

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Divide

Problem 5:
$$ \frac{w^2 x}{y} \div \frac{wx^2}{y^3} $$

1. Flip to multiply:
$$ \frac{w^2 x}{y} \cdot \frac{y^3}{wx^2} $$
2. Simplify variables:
* $w^2 / w = w$ (on top)
* $x / x^2 = 1/x$ ($x$ stays on bottom)
* $y^3 / y = y^2$ (on top)
3. Combine:
$$ \frac{w y^2}{x} $$

Answer: $\frac{wy^2}{x}$

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Problem 6:
$$ \frac{25y^2 - 4}{y^2 - 9} \div \frac{5y - 2}{y + 3} $$

1. Flip to multiply:
$$ \frac{25y^2 - 4}{y^2 - 9} \cdot \frac{y + 3}{5y - 2} $$
2. Factor:
* Top left: $(5y - 2)(5y + 2)$ (Difference of squares)
* Bottom left: $(y - 3)(y + 3)$
3. Cancel:
* $(5y - 2)$ on top left and bottom right cancel.
* $(y + 3)$ on top right and bottom left cancel.
4. Remaining:
* Numerator: $5y + 2$
* Denominator: $y - 3$

Answer: $\frac{5y + 2}{y - 3}$

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Problem 7:
$$ \frac{r^2 v}{t} \div \frac{v^2}{t^2} $$

1. Flip to multiply:
$$ \frac{r^2 v}{t} \cdot \frac{t^2}{v^2} $$
2. Simplify:
* $t^2 / t = t$ (on top)
* $v / v^2 = 1/v$ ($v$ stays on bottom)
3. Combine:
$$ \frac{r^2 t}{v} $$

Answer: $\frac{r^2 t}{v}$

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Problem 8:
$$ \frac{m^2 n}{p} \div \frac{mn^2}{p^2} $$

1. Flip to multiply:
$$ \frac{m^2 n}{p} \cdot \frac{p^2}{mn^2} $$
2. Simplify:
* $m^2 / m = m$ (on top)
* $n / n^2 = 1/n$ ($n$ stays on bottom)
* $p^2 / p = p$ (on top)
3. Combine:
$$ \frac{mp}{n} $$

Answer: $\frac{mp}{n}$

Final Answer:
1. $\frac{42}{5x - 30}$
2. $\frac{(x + 4)^2(x - 3)}{x^3}$
3. $\frac{25t^2}{12(t^2 - 2)(t - 2)}$
4. $\frac{mn(m - n)}{4(m + n)}$
5. $\frac{wy^2}{x}$
6. $\frac{5y + 2}{y - 3}$
7. $\frac{r^2 t}{v}$
8. $\frac{mp}{n}$