15.4E: Exercises
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Simplify a Complex Rational Expression by Writing it as Division
In the following exercises, simplify each complex rational expression by writing it as division.
1. \(\dfrac{\dfrac{2 a}{a+4}}{\dfrac{4 a^{2}}{a^{2}-16}}\)
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\(\dfrac{a-4}{2 a}\)
2. \(\dfrac{\dfrac{3 b}{b-5}}{\dfrac{b^{2}}{b^{2}-25}}\)
3. \(\dfrac{\dfrac{5}{c^{2}+5 c-14}}{\dfrac{10}{c+7}}\)
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\(\dfrac{1}{2(c-2)}\)
4. \(\dfrac{\dfrac{8}{d^{2}+9 d+18}}{\dfrac{12}{d+6}}\)
5. \(\dfrac{\dfrac{1}{2}+\dfrac{5}{6}}{\dfrac{2}{3}+\dfrac{7}{9}}\)
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\(\dfrac{12}{13}\)
6. \(\dfrac{\dfrac{1}{2}+\dfrac{3}{4}}{\dfrac{3}{5}+\dfrac{7}{10}}\)
7. \(\dfrac{\dfrac{2}{3}-\dfrac{1}{9}}{\dfrac{3}{4}+\dfrac{5}{6}}\)
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\(\dfrac{20}{57}\)
8. \(\dfrac{\dfrac{1}{2}-\dfrac{1}{6}}{\dfrac{2}{3}+\dfrac{3}{4}}\)
9. \(\dfrac{\dfrac{n}{m}+\dfrac{1}{n}}{\dfrac{1}{n}-\dfrac{n}{m}}\)
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\(\dfrac{n^{2}+m}{m-n^{2}}\)
10. \(\dfrac{\dfrac{1}{p}+\dfrac{p}{q}}{\dfrac{q}{p}-\dfrac{1}{q}}\)
11. \(\dfrac{\dfrac{1}{r}+\dfrac{1}{t}}{\dfrac{1}{r^{2}}-\dfrac{1}{t^{2}}}\)
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\(\dfrac{r t}{t-r}\)
12. \(\dfrac{\dfrac{2}{v}+\dfrac{2}{w}}{\dfrac{1}{v^{2}}-\dfrac{1}{w^{2}}}\)
13. \(\dfrac{x-\dfrac{2 x}{x+3}}{\dfrac{1}{x+3}+\dfrac{1}{x-3}}\)
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\(\dfrac{(x+1)(x-3)}{2}\)
14. \(\dfrac{y-\dfrac{2 y}{y-4}}{\dfrac{2}{y-4}+\dfrac{2}{y+4}}\)
15. \(\dfrac{2-\dfrac{2}{a+3}}{\dfrac{1}{a+3}+\dfrac{a}{2}}\)
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\(\dfrac{4}{a+1}\)
16. \(\dfrac{4+\dfrac{4}{b-5}}{\dfrac{1}{b-5}+\dfrac{b}{4}}\)
Simplify a Complex Rational Expression by Using the LCD
In the following exercises, simplify each complex rational expression by using the LCD.
17. \(\dfrac{\dfrac{1}{3}+\dfrac{1}{8}}{\dfrac{1}{4}+\dfrac{1}{12}}\)
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\(\dfrac{11}{8}\)
18. \(\dfrac{\dfrac{1}{4}+\dfrac{1}{9}}{\dfrac{1}{6}+\dfrac{1}{12}}\)
19. \(\dfrac{\dfrac{5}{6}+\dfrac{2}{9}}{\dfrac{7}{18}-\dfrac{1}{3}}\)
- Answer
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\(19\)
20. \(\dfrac{\dfrac{1}{6}+\dfrac{4}{15}}{\dfrac{3}{5}-\dfrac{1}{2}}\)
21. \(\dfrac{\dfrac{c}{d}+\dfrac{1}{d}}{\dfrac{1}{d}-\dfrac{d}{c}}\)
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\(\dfrac{c^{2}+c}{c-d^{2}}\)
22. \(\dfrac{\dfrac{1}{m}+\dfrac{m}{n}}{\dfrac{n}{m}-\dfrac{1}{n}}\)
23. \(\dfrac{\dfrac{1}{p}+\dfrac{1}{q}}{\dfrac{1}{p^{2}}-\dfrac{1}{q^{2}}}\)
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\(\dfrac{p q}{q-p}\)
24. \(\dfrac{\dfrac{2}{r}+\dfrac{2}{t}}{\dfrac{1}{r^{2}}-\dfrac{1}{t^{2}}}\)
25. \(\dfrac{\dfrac{2}{x+5}}{\dfrac{3}{x-5}+\dfrac{1}{x^{2}-25}}\)
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\(\dfrac{2 x-10}{3 x+16}\)
26. \(\dfrac{\dfrac{5}{y-4}}{\dfrac{3}{y+4}+\dfrac{2}{y^{2}-16}}\)
27. \(\dfrac{\dfrac{5}{z^{2}-64}+\dfrac{3}{z+8}}{\dfrac{1}{z+8}+\dfrac{2}{z-8}}\)
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\(\dfrac{3 z-19}{3 z+8}\)
28. \(\dfrac{\dfrac{3}{s+6}+\dfrac{5}{s-6}}{\dfrac{1}{s^{2}-36}+\dfrac{4}{s+6}}\)
29. \(\dfrac{\dfrac{4}{a^{2}-2 a-15}}{\dfrac{1}{a-5}+\dfrac{2}{a+3}}\)
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\(\dfrac{4}{3 a-7}\)
30. \(\dfrac{\dfrac{5}{b^{2}-6 b-27}}{\dfrac{3}{b-9}+\dfrac{1}{b+3}}\)
31. \(\dfrac{\dfrac{5}{c+2}-\dfrac{3}{c+7}}{\dfrac{5 c}{c^{2}+9 c+14}}\)
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\(\dfrac{2 c+29}{5 c}\)
32. \(\dfrac{\dfrac{6}{d-4}-\dfrac{2}{d+7}}{\dfrac{2 d}{d^{2}+3 d-28}}\)
33. \(\dfrac{2+\dfrac{1}{p-3}}{\dfrac{5}{p-3}}\)
- Answer
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\(\dfrac{2 p-5}{5}\)
34. \(\dfrac{\dfrac{n}{n-2}}{3+\dfrac{5}{n-2}}\)
35. \(\dfrac{\dfrac{m}{m+5}}{4+\dfrac{1}{m-5}}\)
- Answer
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\(\dfrac{m(m-5)}{(4 m-19)(m+5)}\)
36. \(\dfrac{7+\dfrac{2}{q-2}}{\dfrac{1}{q+2}}\)
In the following exercises, simplify each complex rational expression using either method.
37. \(\dfrac{\dfrac{3}{4}-\dfrac{2}{7}}{\dfrac{1}{2}+\dfrac{5}{14}}\)
- Answer
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\(\dfrac{13}{24}\)
38. \(\dfrac{\dfrac{v}{w}+\dfrac{1}{v}}{\dfrac{1}{v}-\dfrac{v}{w}}\)
39. \(\dfrac{\dfrac{2}{a+4}}{\dfrac{1}{a^{2}-16}}\)
- Answer
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\(2(a-4)\)
40. \(\dfrac{\dfrac{3}{b^{2}-3 b-40}}{\dfrac{5}{b+5}-\dfrac{2}{b-8}}\)
41. \(\dfrac{\dfrac{3}{m}+\dfrac{3}{n}}{\dfrac{1}{m^{2}}-\dfrac{1}{n^{2}}}\)
- Answer
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\(\dfrac{3 m n}{n-m}\)
42. \(\dfrac{\dfrac{2}{r-9}}{\dfrac{1}{r+9}+\dfrac{3}{r^{2}-81}}\)
43. \(\dfrac{x-\dfrac{3 x}{x+2}}{\dfrac{3}{x+2}+\dfrac{3}{x-2}}\)
- Answer
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\(\dfrac{(x-1)(x-2)}{6}\)
44. \(\dfrac{\dfrac{y}{y+3}}{2+\dfrac{1}{y-3}}\)
Writing Exercises
45. In this section, you learned to simplify the complex fraction \(\dfrac{\dfrac{3}{x+2}}{\dfrac{x}{x^{2}-4}}\) two ways: rewriting it as a division problem or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?
- Answer
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Answers will vary.
44. Efraim wants to start simplifying the complex fraction \(\dfrac{\dfrac{1}{a}+\dfrac{1}{b}}{\dfrac{1}{a}-\dfrac{1}{b}}\) by cancelling the variables from the numerator and denominator, \(\dfrac{\dfrac{1}{\cancel{a}}+\dfrac{1}{\cancel {b}}}{\dfrac{1}{\cancel{a}}-\dfrac{1}{\cancel{b}}}\). Explain what is wrong with Efraim’s plan.