Adding And Subtracting Rational Expressions Worksheet - Free Printable
Educational worksheet: Adding And Subtracting Rational Expressions Worksheet. Download and print for classroom or home learning activities.
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Step-by-step solution for: Adding And Subtracting Rational Expressions Worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Adding And Subtracting Rational Expressions Worksheet
Since all options (ExplainAnswer, TeachTopic, GeneratePractice) are false, I will only provide the final answers for each problem, numbered 13 through 42, simplified as much as possible.
Let me solve each one step by step carefully:
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13. $\frac{a}{y} \cdot \frac{x}{a} = \frac{a x}{y a} = \frac{x}{y}$
→ $\boxed{\dfrac{x}{y}}$
14. $\frac{r}{s} \div \frac{t}{s} = \frac{r}{s} \cdot \frac{s}{t} = \frac{r s}{s t} = \frac{r}{t}$
→ $\boxed{\dfrac{r}{t}}$
15. $\frac{2x}{3y} \cdot \frac{y^2}{4x} = \frac{2x y^2}{3y \cdot 4x} = \frac{2 y}{12} = \frac{y}{6}$
(Cancel $x$, and $y^2 / y = y$; $2/12 = 1/6$)
→ $\boxed{\dfrac{y}{6}}$
16. $\frac{14b}{9a} \div \frac{7a}{3} = \frac{14b}{9a} \cdot \frac{3}{7a} = \frac{14b \cdot 3}{9a \cdot 7a} = \frac{42b}{63a^2} = \frac{2b}{3a^2}$
(Simplify: $42/63 = 2/3$)
→ $\boxed{\dfrac{2b}{3a^2}}$
17. $\frac{12(t-1)}{t^2+1} \div \frac{t^2-1}{t+1} = \frac{12(t-1)}{t^2+1} \cdot \frac{t+1}{t^2 - 1}$
Note: $t^2 - 1 = (t-1)(t+1)$
So: $\frac{12(t-1)(t+1)}{(t^2+1)(t-1)(t+1)} = \frac{12}{t^2 + 1}$
(Cancel $(t-1)(t+1)$)
→ $\boxed{\dfrac{12}{t^2 + 1}}$
18. $\frac{3x^2 + 2x}{x - 2} \cdot \frac{x - 2}{x} = \frac{x(3x + 2)}{x - 2} \cdot \frac{x - 2}{x} = 3x + 2$
(Cancel $x$ and $x-2$)
→ $\boxed{3x + 2}$
19. $\frac{6y - 2}{y} \cdot \frac{6y}{6y + 3} = \frac{2(3y - 1)}{y} \cdot \frac{6y}{3(2y + 1)} = \frac{2(3y - 1) \cdot 6y}{y \cdot 3(2y + 1)}$
Simplify: cancel $y$, $6/3 = 2$:
= $\frac{2(3y - 1) \cdot 2}{2y + 1} = \frac{4(3y - 1)}{2y + 1} = \frac{12y - 4}{2y + 1}$
→ $\boxed{\dfrac{12y - 4}{2y + 1}}$
20. $\frac{t + 2}{t^2 + 3t} \div \frac{t + 2}{t + 3} = \frac{t+2}{t(t+3)} \cdot \frac{t+3}{t+2} = \frac{1}{t}$
(Cancel $t+2$, $t+3$)
→ $\boxed{\dfrac{1}{t}}$
21. $\frac{4k - 8}{k + 1} \div \frac{2k - 10}{k + 1} = \frac{4(k - 2)}{k + 1} \cdot \frac{k + 1}{2(k - 5)} = \frac{4(k - 2)}{2(k - 5)} = \frac{2(k - 2)}{k - 5}$
→ $\boxed{\dfrac{2(k - 2)}{k - 5}}$
22. $\frac{5x^2 + 2x}{x - 2} \div \frac{5x + 2}{4x - 8} = \frac{x(5x + 2)}{x - 2} \cdot \frac{4(x - 2)}{5x + 2} = x \cdot 4 = 4x$
(Cancel $5x+2$, $x-2$)
→ $\boxed{4x}$
23. $\frac{a^2 + ab}{b} \cdot \frac{b}{a^2 - ab} = \frac{a(a + b)}{b} \cdot \frac{b}{a(a - b)} = \frac{a + b}{a - b}$
(Cancel $a$, $b$)
→ $\boxed{\dfrac{a + b}{a - b}}$
24. $\frac{10}{x + 2} \div \frac{x^2 + 2x}{4x + 6} = \frac{10}{x+2} \cdot \frac{4x + 6}{x(x + 2)} = \frac{10(2)(2x + 3)}{(x+2) \cdot x(x+2)} = \frac{20(2x + 3)}{x(x+2)^2}$
Wait — better factor:
$4x + 6 = 2(2x + 3)$, $x^2 + 2x = x(x+2)$
So:
$\frac{10}{x+2} \cdot \frac{2(2x+3)}{x(x+2)} = \frac{20(2x+3)}{x(x+2)^2}$
No further simplification.
→ $\boxed{\dfrac{20(2x + 3)}{x(x + 2)^2}}$
But maybe we can leave factored: $\frac{20(2x+3)}{x(x+2)^2}$ is simplest.
25. $\frac{9x^2 - 4}{27x^3 - 8} \div \frac{3x + 2}{3x - 2}$
Note: $9x^2 - 4 = (3x - 2)(3x + 2)$
$27x^3 - 8 = (3x)^3 - 2^3 = (3x - 2)(9x^2 + 6x + 4)$
So expression becomes:
$\frac{(3x - 2)(3x + 2)}{(3x - 2)(9x^2 + 6x + 4)} \cdot \frac{3x - 2}{3x + 2} = \frac{3x - 2}{9x^2 + 6x + 4}$
(Cancel $(3x-2)$ once, $(3x+2)$)
→ $\boxed{\dfrac{3x - 2}{9x^2 + 6x + 4}}$
26. $\frac{x^2 - x}{4x^2 - 9} \div \frac{x - 1}{2x + 3} = \frac{x(x - 1)}{(2x - 3)(2x + 3)} \cdot \frac{2x + 3}{x - 1} = \frac{x}{2x - 3}$
(Cancel $x-1$, $2x+3$)
→ $\boxed{\dfrac{x}{2x - 3}}$
27. $\frac{x^2 + 10x + 25}{x - 4} \cdot \frac{3x - 12}{2x + 10} = \frac{(x+5)^2}{x - 4} \cdot \frac{3(x - 4)}{2(x + 5)} = \frac{(x+5)^2 \cdot 3(x - 4)}{(x - 4) \cdot 2(x + 5)} = \frac{3(x + 5)}{2}$
→ $\boxed{\dfrac{3(x + 5)}{2}}$
28. $\frac{3}{a} + \frac{2}{b} = \frac{3b + 2a}{ab}$
→ $\boxed{\dfrac{3b + 2a}{ab}}$
29. $\frac{5}{x + 2} + \frac{x}{x + 2} = \frac{5 + x}{x + 2} = \frac{x + 5}{x + 2}$
→ $\boxed{\dfrac{x + 5}{x + 2}}$
30. $\frac{9}{2x + 1} - \frac{5}{2x + 1} = \frac{4}{2x + 1}$
→ $\boxed{\dfrac{4}{2x + 1}}$
31. $\frac{9}{r} + \frac{8}{r} = \frac{17}{r}$
→ $\boxed{\dfrac{17}{r}}$
32. $\frac{x}{z} + \frac{z}{y} = \frac{xy + z^2}{zy}$
(LCM = $zy$)
→ $\boxed{\dfrac{xy + z^2}{yz}}$
33. $\frac{5}{xy} - 3 = \frac{5 - 3xy}{xy}$
→ $\boxed{\dfrac{5 - 3xy}{xy}}$
34. $\frac{8}{m + 3} + \frac{2}{m} = \frac{8m + 2(m + 3)}{m(m + 3)} = \frac{8m + 2m + 6}{m(m+3)} = \frac{10m + 6}{m(m+3)} = \frac{2(5m + 3)}{m(m+3)}$
→ $\boxed{\dfrac{2(5m + 3)}{m(m + 3)}}$
35. $\frac{b}{b - 2} + \frac{4}{b - 3} = \frac{b(b - 3) + 4(b - 2)}{(b - 2)(b - 3)} = \frac{b^2 - 3b + 4b - 8}{(b-2)(b-3)} = \frac{b^2 + b - 8}{(b - 2)(b - 3)}$
→ $\boxed{\dfrac{b^2 + b - 8}{(b - 2)(b - 3)}}$
36. $\frac{n}{n + 3} + \frac{3}{n + 3} = \frac{n + 3}{n + 3} = 1$
→ $\boxed{1}$
37. $\frac{z}{z + 2} - \frac{4}{z - 2} = \frac{z(z - 2) - 4(z + 2)}{(z + 2)(z - 2)} = \frac{z^2 - 2z - 4z - 8}{z^2 - 4} = \frac{z^2 - 6z - 8}{z^2 - 4}$
→ $\boxed{\dfrac{z^2 - 6z - 8}{z^2 - 4}}$
38. $\frac{10}{x - y} - \frac{2}{y - x} = \frac{10}{x - y} + \frac{2}{x - y} = \frac{12}{x - y}$
(since $y - x = -(x - y)$)
→ $\boxed{\dfrac{12}{x - y}}$
39. $\frac{3}{x} + \frac{2}{x - 2} = \frac{3(x - 2) + 2x}{x(x - 2)} = \frac{3x - 6 + 2x}{x(x - 2)} = \frac{5x - 6}{x(x - 2)}$
→ $\boxed{\dfrac{5x - 6}{x(x - 2)}}$
40. $\frac{t}{t + 2} + \frac{2}{t - 1} = \frac{t(t - 1) + 2(t + 2)}{(t + 2)(t - 1)} = \frac{t^2 - t + 2t + 4}{(t+2)(t-1)} = \frac{t^2 + t + 4}{(t + 2)(t - 1)}$
→ $\boxed{\dfrac{t^2 + t + 4}{(t + 2)(t - 1)}}$
41. $\frac{2x}{x - 1} - \frac{5}{x + 1} = \frac{2x(x + 1) - 5(x - 1)}{(x - 1)(x + 1)} = \frac{2x^2 + 2x - 5x + 5}{x^2 - 1} = \frac{2x^2 - 3x + 5}{x^2 - 1}$
→ $\boxed{\dfrac{2x^2 - 3x + 5}{x^2 - 1}}$
42. $\frac{y + 2}{y^2 - 4} + \frac{y^2 + 2y + 4}{y^3 - 8}$
Factor:
$y^2 - 4 = (y - 2)(y + 2)$
$y^3 - 8 = (y - 2)(y^2 + 2y + 4)$
So:
$\frac{y + 2}{(y - 2)(y + 2)} + \frac{y^2 + 2y + 4}{(y - 2)(y^2 + 2y + 4)} = \frac{1}{y - 2} + \frac{1}{y - 2} = \frac{2}{y - 2}$
→ $\boxed{\dfrac{2}{y - 2}}$
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Final Answer:
13. $\dfrac{x}{y}$
14. $\dfrac{r}{t}$
15. $\dfrac{y}{6}$
16. $\dfrac{2b}{3a^2}$
17. $\dfrac{12}{t^2 + 1}$
18. $3x + 2$
19. $\dfrac{12y - 4}{2y + 1}$
20. $\dfrac{1}{t}$
21. $\dfrac{2(k - 2)}{k - 5}$
22. $4x$
23. $\dfrac{a + b}{a - b}$
24. $\dfrac{20(2x + 3)}{x(x + 2)^2}$
25. $\dfrac{3x - 2}{9x^2 + 6x + 4}$
26. $\dfrac{x}{2x - 3}$
27. $\dfrac{3(x + 5)}{2}$
28. $\dfrac{3b + 2a}{ab}$
29. $\dfrac{x + 5}{x + 2}$
30. $\dfrac{4}{2x + 1}$
31. $\dfrac{17}{r}$
32. $\dfrac{xy + z^2}{yz}$
33. $\dfrac{5 - 3xy}{xy}$
34. $\dfrac{2(5m + 3)}{m(m + 3)}$
35. $\dfrac{b^2 + b - 8}{(b - 2)(b - 3)}$
36. $1$
37. $\dfrac{z^2 - 6z - 8}{z^2 - 4}$
38. $\dfrac{12}{x - y}$
39. $\dfrac{5x - 6}{x(x - 2)}$
40. $\dfrac{t^2 + t + 4}{(t + 2)(t - 1)}$
41. $\dfrac{2x^2 - 3x + 5}{x^2 - 1}$
42. $\dfrac{2}{y - 2}$
Let me solve each one step by step carefully:
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13. $\frac{a}{y} \cdot \frac{x}{a} = \frac{a x}{y a} = \frac{x}{y}$
→ $\boxed{\dfrac{x}{y}}$
14. $\frac{r}{s} \div \frac{t}{s} = \frac{r}{s} \cdot \frac{s}{t} = \frac{r s}{s t} = \frac{r}{t}$
→ $\boxed{\dfrac{r}{t}}$
15. $\frac{2x}{3y} \cdot \frac{y^2}{4x} = \frac{2x y^2}{3y \cdot 4x} = \frac{2 y}{12} = \frac{y}{6}$
(Cancel $x$, and $y^2 / y = y$; $2/12 = 1/6$)
→ $\boxed{\dfrac{y}{6}}$
16. $\frac{14b}{9a} \div \frac{7a}{3} = \frac{14b}{9a} \cdot \frac{3}{7a} = \frac{14b \cdot 3}{9a \cdot 7a} = \frac{42b}{63a^2} = \frac{2b}{3a^2}$
(Simplify: $42/63 = 2/3$)
→ $\boxed{\dfrac{2b}{3a^2}}$
17. $\frac{12(t-1)}{t^2+1} \div \frac{t^2-1}{t+1} = \frac{12(t-1)}{t^2+1} \cdot \frac{t+1}{t^2 - 1}$
Note: $t^2 - 1 = (t-1)(t+1)$
So: $\frac{12(t-1)(t+1)}{(t^2+1)(t-1)(t+1)} = \frac{12}{t^2 + 1}$
(Cancel $(t-1)(t+1)$)
→ $\boxed{\dfrac{12}{t^2 + 1}}$
18. $\frac{3x^2 + 2x}{x - 2} \cdot \frac{x - 2}{x} = \frac{x(3x + 2)}{x - 2} \cdot \frac{x - 2}{x} = 3x + 2$
(Cancel $x$ and $x-2$)
→ $\boxed{3x + 2}$
19. $\frac{6y - 2}{y} \cdot \frac{6y}{6y + 3} = \frac{2(3y - 1)}{y} \cdot \frac{6y}{3(2y + 1)} = \frac{2(3y - 1) \cdot 6y}{y \cdot 3(2y + 1)}$
Simplify: cancel $y$, $6/3 = 2$:
= $\frac{2(3y - 1) \cdot 2}{2y + 1} = \frac{4(3y - 1)}{2y + 1} = \frac{12y - 4}{2y + 1}$
→ $\boxed{\dfrac{12y - 4}{2y + 1}}$
20. $\frac{t + 2}{t^2 + 3t} \div \frac{t + 2}{t + 3} = \frac{t+2}{t(t+3)} \cdot \frac{t+3}{t+2} = \frac{1}{t}$
(Cancel $t+2$, $t+3$)
→ $\boxed{\dfrac{1}{t}}$
21. $\frac{4k - 8}{k + 1} \div \frac{2k - 10}{k + 1} = \frac{4(k - 2)}{k + 1} \cdot \frac{k + 1}{2(k - 5)} = \frac{4(k - 2)}{2(k - 5)} = \frac{2(k - 2)}{k - 5}$
→ $\boxed{\dfrac{2(k - 2)}{k - 5}}$
22. $\frac{5x^2 + 2x}{x - 2} \div \frac{5x + 2}{4x - 8} = \frac{x(5x + 2)}{x - 2} \cdot \frac{4(x - 2)}{5x + 2} = x \cdot 4 = 4x$
(Cancel $5x+2$, $x-2$)
→ $\boxed{4x}$
23. $\frac{a^2 + ab}{b} \cdot \frac{b}{a^2 - ab} = \frac{a(a + b)}{b} \cdot \frac{b}{a(a - b)} = \frac{a + b}{a - b}$
(Cancel $a$, $b$)
→ $\boxed{\dfrac{a + b}{a - b}}$
24. $\frac{10}{x + 2} \div \frac{x^2 + 2x}{4x + 6} = \frac{10}{x+2} \cdot \frac{4x + 6}{x(x + 2)} = \frac{10(2)(2x + 3)}{(x+2) \cdot x(x+2)} = \frac{20(2x + 3)}{x(x+2)^2}$
Wait — better factor:
$4x + 6 = 2(2x + 3)$, $x^2 + 2x = x(x+2)$
So:
$\frac{10}{x+2} \cdot \frac{2(2x+3)}{x(x+2)} = \frac{20(2x+3)}{x(x+2)^2}$
No further simplification.
→ $\boxed{\dfrac{20(2x + 3)}{x(x + 2)^2}}$
But maybe we can leave factored: $\frac{20(2x+3)}{x(x+2)^2}$ is simplest.
25. $\frac{9x^2 - 4}{27x^3 - 8} \div \frac{3x + 2}{3x - 2}$
Note: $9x^2 - 4 = (3x - 2)(3x + 2)$
$27x^3 - 8 = (3x)^3 - 2^3 = (3x - 2)(9x^2 + 6x + 4)$
So expression becomes:
$\frac{(3x - 2)(3x + 2)}{(3x - 2)(9x^2 + 6x + 4)} \cdot \frac{3x - 2}{3x + 2} = \frac{3x - 2}{9x^2 + 6x + 4}$
(Cancel $(3x-2)$ once, $(3x+2)$)
→ $\boxed{\dfrac{3x - 2}{9x^2 + 6x + 4}}$
26. $\frac{x^2 - x}{4x^2 - 9} \div \frac{x - 1}{2x + 3} = \frac{x(x - 1)}{(2x - 3)(2x + 3)} \cdot \frac{2x + 3}{x - 1} = \frac{x}{2x - 3}$
(Cancel $x-1$, $2x+3$)
→ $\boxed{\dfrac{x}{2x - 3}}$
27. $\frac{x^2 + 10x + 25}{x - 4} \cdot \frac{3x - 12}{2x + 10} = \frac{(x+5)^2}{x - 4} \cdot \frac{3(x - 4)}{2(x + 5)} = \frac{(x+5)^2 \cdot 3(x - 4)}{(x - 4) \cdot 2(x + 5)} = \frac{3(x + 5)}{2}$
→ $\boxed{\dfrac{3(x + 5)}{2}}$
28. $\frac{3}{a} + \frac{2}{b} = \frac{3b + 2a}{ab}$
→ $\boxed{\dfrac{3b + 2a}{ab}}$
29. $\frac{5}{x + 2} + \frac{x}{x + 2} = \frac{5 + x}{x + 2} = \frac{x + 5}{x + 2}$
→ $\boxed{\dfrac{x + 5}{x + 2}}$
30. $\frac{9}{2x + 1} - \frac{5}{2x + 1} = \frac{4}{2x + 1}$
→ $\boxed{\dfrac{4}{2x + 1}}$
31. $\frac{9}{r} + \frac{8}{r} = \frac{17}{r}$
→ $\boxed{\dfrac{17}{r}}$
32. $\frac{x}{z} + \frac{z}{y} = \frac{xy + z^2}{zy}$
(LCM = $zy$)
→ $\boxed{\dfrac{xy + z^2}{yz}}$
33. $\frac{5}{xy} - 3 = \frac{5 - 3xy}{xy}$
→ $\boxed{\dfrac{5 - 3xy}{xy}}$
34. $\frac{8}{m + 3} + \frac{2}{m} = \frac{8m + 2(m + 3)}{m(m + 3)} = \frac{8m + 2m + 6}{m(m+3)} = \frac{10m + 6}{m(m+3)} = \frac{2(5m + 3)}{m(m+3)}$
→ $\boxed{\dfrac{2(5m + 3)}{m(m + 3)}}$
35. $\frac{b}{b - 2} + \frac{4}{b - 3} = \frac{b(b - 3) + 4(b - 2)}{(b - 2)(b - 3)} = \frac{b^2 - 3b + 4b - 8}{(b-2)(b-3)} = \frac{b^2 + b - 8}{(b - 2)(b - 3)}$
→ $\boxed{\dfrac{b^2 + b - 8}{(b - 2)(b - 3)}}$
36. $\frac{n}{n + 3} + \frac{3}{n + 3} = \frac{n + 3}{n + 3} = 1$
→ $\boxed{1}$
37. $\frac{z}{z + 2} - \frac{4}{z - 2} = \frac{z(z - 2) - 4(z + 2)}{(z + 2)(z - 2)} = \frac{z^2 - 2z - 4z - 8}{z^2 - 4} = \frac{z^2 - 6z - 8}{z^2 - 4}$
→ $\boxed{\dfrac{z^2 - 6z - 8}{z^2 - 4}}$
38. $\frac{10}{x - y} - \frac{2}{y - x} = \frac{10}{x - y} + \frac{2}{x - y} = \frac{12}{x - y}$
(since $y - x = -(x - y)$)
→ $\boxed{\dfrac{12}{x - y}}$
39. $\frac{3}{x} + \frac{2}{x - 2} = \frac{3(x - 2) + 2x}{x(x - 2)} = \frac{3x - 6 + 2x}{x(x - 2)} = \frac{5x - 6}{x(x - 2)}$
→ $\boxed{\dfrac{5x - 6}{x(x - 2)}}$
40. $\frac{t}{t + 2} + \frac{2}{t - 1} = \frac{t(t - 1) + 2(t + 2)}{(t + 2)(t - 1)} = \frac{t^2 - t + 2t + 4}{(t+2)(t-1)} = \frac{t^2 + t + 4}{(t + 2)(t - 1)}$
→ $\boxed{\dfrac{t^2 + t + 4}{(t + 2)(t - 1)}}$
41. $\frac{2x}{x - 1} - \frac{5}{x + 1} = \frac{2x(x + 1) - 5(x - 1)}{(x - 1)(x + 1)} = \frac{2x^2 + 2x - 5x + 5}{x^2 - 1} = \frac{2x^2 - 3x + 5}{x^2 - 1}$
→ $\boxed{\dfrac{2x^2 - 3x + 5}{x^2 - 1}}$
42. $\frac{y + 2}{y^2 - 4} + \frac{y^2 + 2y + 4}{y^3 - 8}$
Factor:
$y^2 - 4 = (y - 2)(y + 2)$
$y^3 - 8 = (y - 2)(y^2 + 2y + 4)$
So:
$\frac{y + 2}{(y - 2)(y + 2)} + \frac{y^2 + 2y + 4}{(y - 2)(y^2 + 2y + 4)} = \frac{1}{y - 2} + \frac{1}{y - 2} = \frac{2}{y - 2}$
→ $\boxed{\dfrac{2}{y - 2}}$
---
Final Answer:
13. $\dfrac{x}{y}$
14. $\dfrac{r}{t}$
15. $\dfrac{y}{6}$
16. $\dfrac{2b}{3a^2}$
17. $\dfrac{12}{t^2 + 1}$
18. $3x + 2$
19. $\dfrac{12y - 4}{2y + 1}$
20. $\dfrac{1}{t}$
21. $\dfrac{2(k - 2)}{k - 5}$
22. $4x$
23. $\dfrac{a + b}{a - b}$
24. $\dfrac{20(2x + 3)}{x(x + 2)^2}$
25. $\dfrac{3x - 2}{9x^2 + 6x + 4}$
26. $\dfrac{x}{2x - 3}$
27. $\dfrac{3(x + 5)}{2}$
28. $\dfrac{3b + 2a}{ab}$
29. $\dfrac{x + 5}{x + 2}$
30. $\dfrac{4}{2x + 1}$
31. $\dfrac{17}{r}$
32. $\dfrac{xy + z^2}{yz}$
33. $\dfrac{5 - 3xy}{xy}$
34. $\dfrac{2(5m + 3)}{m(m + 3)}$
35. $\dfrac{b^2 + b - 8}{(b - 2)(b - 3)}$
36. $1$
37. $\dfrac{z^2 - 6z - 8}{z^2 - 4}$
38. $\dfrac{12}{x - y}$
39. $\dfrac{5x - 6}{x(x - 2)}$
40. $\dfrac{t^2 + t + 4}{(t + 2)(t - 1)}$
41. $\dfrac{2x^2 - 3x + 5}{x^2 - 1}$
42. $\dfrac{2}{y - 2}$
Parent Tip: Review the logic above to help your child master the concept of operations with rational expressions worksheet.
