Worksheet on Operations on Rational Algebraic Expressions | PDF - Free Printable

Here are the step-by-step solutions for each problem on the worksheet. The main rule for all these problems is to factor everything first, then cancel out common terms, and finally combine what is left.

1. $\frac{8n}{4n + 4} \cdot \frac{8n + 8}{10}$

* Step 1: Factor the denominators and numerators where possible.
* $4n + 4$ becomes $4(n + 1)$.
* $8n + 8$ becomes $8(n + 1)$.
* The expression is now: $\frac{8n}{4(n + 1)} \cdot \frac{8(n + 1)}{10}$
* Step 2: Cancel common terms.
* The $(n + 1)$ on the bottom left cancels with the $(n + 1)$ on the top right.
* We are left with numbers: $\frac{8n \cdot 8}{4 \cdot 10} = \frac{64n}{40}$
* Step 3: Simplify the fraction $\frac{64}{40}$. Both are divisible by 8.
* $64 \div 8 = 8$
* $40 \div 8 = 5$
* Result: $\frac{8n}{5}$

2. $\frac{15p + 30}{p + 2} \cdot \frac{p + 3}{15p + 45}$

* Step 1: Factor.
* $15p + 30$ becomes $15(p + 2)$.
* $15p + 45$ becomes $15(p + 3)$.
* Expression: $\frac{15(p + 2)}{p + 2} \cdot \frac{p + 3}{15(p + 3)}$
* Step 2: Cancel terms.
* $(p + 2)$ cancels with $(p + 2)$.
* $(p + 3)$ cancels with $(p + 3)$.
* $15$ on top cancels with $15$ on bottom.
* Result: $1$

3. $\frac{m^2 - n^2}{m} \cdot \frac{n}{3m + 3n}$

* Step 1: Factor.
* $m^2 - n^2$ is a difference of squares: $(m - n)(m + n)$.
* $3m + 3n$ becomes $3(m + n)$.
* Expression: $\frac{(m - n)(m + n)}{m} \cdot \frac{n}{3(m + n)}$
* Step 2: Cancel terms.
* $(m + n)$ on top cancels with $(m + n)$ on bottom.
* Step 3: Multiply remaining parts.
* Top: $n(m - n)$ or $mn - n^2$
* Bottom: $3m$
* Result: $\frac{n(m - n)}{3m}$

4. $\frac{1}{v - 4} \cdot \frac{v^2 - 7v + 12}{v + 3}$

* Step 1: Factor the quadratic equation $v^2 - 7v + 12$.
* We need two numbers that multiply to 12 and add to -7. Those are -3 and -4.
* So, $v^2 - 7v + 12 = (v - 3)(v - 4)$.
* Expression: $\frac{1}{v - 4} \cdot \frac{(v - 3)(v - 4)}{v + 3}$
* Step 2: Cancel terms.
* $(v - 4)$ on the bottom left cancels with $(v - 4)$ on the top right.
* Step 3: Combine remaining parts.
* Top: $1 \cdot (v - 3) = v - 3$
* Bottom: $v + 3$
* Result: $\frac{v - 3}{v + 3}$

5. $\frac{1}{x - 5} \div \frac{5}{6x + 30}$

* Step 1: Change division to multiplication by flipping the second fraction (reciprocal).
* $\frac{1}{x - 5} \cdot \frac{6x + 30}{5}$
* Step 2: Factor.
* $6x + 30$ becomes $6(x + 5)$.
* Expression: $\frac{1}{x - 5} \cdot \frac{6(x + 5)}{5}$
* Step 3: Check for cancellations.
* There are no common terms to cancel ($x-5$ is not the same as $x+5$).
* Step 4: Multiply across.
* Top: $1 \cdot 6(x + 5) = 6(x + 5)$
* Bottom: $(x - 5) \cdot 5 = 5(x - 5)$
* Result: $\frac{6(x + 5)}{5(x - 5)}$

6. $\frac{p + 5}{8} \div \frac{2p + 10}{8p + 8}$

* Step 1: Flip the second fraction to multiply.
* $\frac{p + 5}{8} \cdot \frac{8p + 8}{2p + 10}$
* Step 2: Factor.
* $8p + 8$ becomes $8(p + 1)$.
* $2p + 10$ becomes $2(p + 5)$.
* Expression: $\frac{p + 5}{8} \cdot \frac{8(p + 1)}{2(p + 5)}$
* Step 3: Cancel terms.
* $(p + 5)$ cancels with $(p + 5)$.
* $8$ on the bottom left cancels with $8$ on the top right.
* Step 4: Remaining parts.
* Top: $p + 1$
* Bottom: $2$
* Result: $\frac{p + 1}{2}$

7. $\frac{5n}{30m} + \frac{2m + 2n}{30m}$

* Step 1: Since the denominators are the same ($30m$), just add the numerators.
* $\frac{5n + 2m + 2n}{30m}$
* Step 2: Combine like terms in the numerator ($5n + 2n = 7n$).
* $\frac{2m + 7n}{30m}$
* Step 3: Check if it simplifies. No common factor for the whole numerator and denominator.
* Result: $\frac{2m + 7n}{30m}$

8. $\frac{x + y}{18xy} - \frac{6x + y}{18xy}$

* Step 1: Denominators are the same, so subtract the numerators. Be careful with the minus sign!
* $\frac{(x + y) - (6x + y)}{18xy}$
* Step 2: Distribute the negative sign.
* $x + y - 6x - y$
* Step 3: Combine like terms.
* $x - 6x = -5x$
* $y - y = 0$
* Numerator is $-5x$.
* Expression: $\frac{-5x}{18xy}$
* Step 4: Simplify.
* $x$ on top cancels with $x$ on bottom.
* $\frac{-5}{18y}$
* Result: $-\frac{5}{18y}$

9. $\frac{m^2 - n^2}{m} - \frac{n}{3m + 3n}$

* Step 1: Find a Common Denominator (LCD).
* First term denominator: $m$
* Second term denominator: $3(m + n)$ (factored)
* LCD is $3m(m + n)$.
* Step 2: Adjust fractions to have the LCD.
* First term: Multiply top and bottom by $3(m + n)$.
* $\frac{3(m + n)(m^2 - n^2)}{3m(m + n)}$
* Second term: Multiply top and bottom by $m$.
* $\frac{mn}{3m(m + n)}$
* Step 3: Expand the first numerator.
* Recall $m^2 - n^2 = (m - n)(m + n)$.
* Numerator 1: $3(m + n)(m - n)(m + n) = 3(m - n)(m + n)^2$. This looks complicated. Let's try expanding directly:
* $3(m+n)(m^2-n^2) = 3(m^3 - mn^2 + m^2n - n^3) = 3m^3 - 3mn^2 + 3m^2n - 3n^3$.
* Numerator 2: $mn$.
* Step 4: Subtract numerators.
* $(3m^3 - 3mn^2 + 3m^2n - 3n^3) - (mn)$
* $3m^3 + 3m^2n - 3mn^2 - mn - 3n^3$
* This does not simplify easily. Let's re-evaluate if there was a simpler cancellation intended. Usually, these worksheets have clean answers.
* Let's check the subtraction again: $\frac{m^2-n^2}{m} - \frac{n}{3(m+n)}$.
* Common Denom: $3m(m+n)$.
* Top: $3(m+n)(m^2-n^2) - mn$.
* $3(m+n)(m-n)(m+n) - mn = 3(m-n)(m+n)^2 - mn$.
* This is the exact form. It doesn't reduce further nicely.
* Alternative interpretation: Did I misread the question? No, it looks correct.
* Let's leave it in factored form for clarity:
* $\frac{3(m+n)(m^2-n^2) - mn}{3m(m+n)}$
* Or expanded: $\frac{3m^3 + 3m^2n - 3mn^2 - 3n^3 - mn}{3m(m+n)}$

*(Self-Correction/Refinement for #9)*: Often in these specific algebra sheets, if the answer is messy, double check for simple errors.
Let's look at Problem 10 and 11 to ensure the pattern holds. They seem standard. Problem 9 is indeed the most complex one. I will provide the combined single fraction.

10. $\frac{5x - 4}{9x^2 + 27x^2} - \frac{x + 6}{9x^2 + 27x^2}$

* Note: The denominator in the image says $9x^2 + 27x^2$. This combines to $36x^2$. However, usually, these problems involve factoring like $9x^2 + 27x$. Let's look closely at the image.
* Image text: $9x^2 + 27x^2$. Wait, looking at problem 10, the denominator is written as $9x^2 + 27x^2$? No, typically it is $9x^2 + 27x$. Let me zoom in mentally.
* Actually, looking at the spacing, it is likely $9x^2 + 27x$. If it were $x^2$ twice, they would just write $36x^2$. The variable change suggests a typo in my initial read or the sheet. Let's assume the standard form $9x^2 + 27x$ which factors to $9x(x+3)$.
* Let's check the second denominator: $9x^2 + 27x^2$? No, the second one is also $9x^2 + 27x^2$?
* Let's look at the image again very carefully.
* Problem 10 Denom 1: $9x^2 + 27x^2$ ?? No, the exponent on the second term is hard to see but contextually it is almost certainly $27x$.
* Problem 10 Denom 2: $9x^2 + 27x^2$ ?? Same here.
* Hypothesis: The denominators are identical.
* If Denom is $9x^2 + 27x$: Factors to $9x(x+3)$.
* If Denom is $9x^2 + 27x^2$: It is $36x^2$.
* Let's solve for $9x^2 + 27x$ as it is the standard algebraic structure for these "factor completely" worksheets.
* Denominator: $9x(x + 3)$.
* Numerator: $(5x - 4) - (x + 6) = 5x - 4 - x - 6 = 4x - 10$.
* Expression: $\frac{4x - 10}{9x(x + 3)}$.
* Factor numerator: $2(2x - 5)$.
* Result: $\frac{2(2x - 5)}{9x(x + 3)}$.

* *What if the denominator really is $36x^2$?*
* Num: $4x - 10$.
* $\frac{4x - 10}{36x^2} = \frac{2(2x - 5)}{36x^2} = \frac{2x - 5}{18x^2}$.
* Given the instruction "Factor Completely", the $9x(x+3)$ path is much more likely intended because it tests factoring skills. I will proceed with the assumption that the second term is $27x$.

11. $\frac{5p}{p - 3} + \frac{5}{p + 6}$

* Step 1: Find LCD.
* Denominators are $(p - 3)$ and $(p + 6)$.
* LCD: $(p - 3)(p + 6)$.
* Step 2: Adjust fractions.
* First term: $\frac{5p(p + 6)}{(p - 3)(p + 6)}$
* Second term: $\frac{5(p - 3)}{(p - 3)(p + 6)}$
* Step 3: Expand numerators.
* $5p(p + 6) = 5p^2 + 30p$
* $5(p - 3) = 5p - 15$
* Step 4: Add numerators.
* $5p^2 + 30p + 5p - 15 = 5p^2 + 35p - 15$
* Step 5: Factor the new numerator to see if it simplifies.
* Factor out 5: $5(p^2 + 7p - 3)$.
* Does $p^2 + 7p - 3$ factor further? We need numbers multiplying to -3 adding to 7. No integers work.
* Result: $\frac{5(p^2 + 7p - 3)}{(p - 3)(p + 6)}$

12. $\frac{1}{a^2 + 3a + 2} + \frac{2}{a + 1}$

* Step 1: Factor the first denominator.
* $a^2 + 3a + 2$ factors into $(a + 2)(a + 1)$.
* Second denominator is $(a + 1)$.
* LCD is $(a + 2)(a + 1)$.
* Step 2: Adjust fractions.
* First term stays: $\frac{1}{(a + 2)(a + 1)}$
* Second term needs $(a + 2)$ on top and bottom: $\frac{2(a + 2)}{(a + 1)(a + 2)}$
* Step 3: Add numerators.
* $1 + 2(a + 2) = 1 + 2a + 4 = 2a + 5$
* Step 4: Combine.
* $\frac{2a + 5}{(a + 2)(a + 1)}$
* Result: $\frac{2a + 5}{(a + 2)(a + 1)}$

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Final Answer:

1. $\frac{8n}{5}$
2. $1$
3. $\frac{n(m - n)}{3m}$
4. $\frac{v - 3}{v + 3}$
5. $\frac{6(x + 5)}{5(x - 5)}$
6. $\frac{p + 1}{2}$
7. $\frac{2m + 7n}{30m}$
8. $-\frac{5}{18y}$
9. $\frac{3(m+n)(m^2-n^2) - mn}{3m(m+n)}$ *(Note: This expression does not simplify cleanly)*
10. $\frac{2(2x - 5)}{9x(x + 3)}$ *(Assuming denominator is $9x^2+27x$)*
11. $\frac{5(p^2 + 7p - 3)}{(p - 3)(p + 6)}$
12. $\frac{2a + 5}{(a + 2)(a + 1)}$