7.4E: Exercises

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Jan 27, 2022
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- 7.4: Simplify Complex Rational Expressions
- 7.5: Solve Rational Equations

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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. $\dfrac{\dfrac{2 a}{a+4}}{\dfrac{4 a^{2}}{a^{2}-16}}$

Answer: $\dfrac{a-4}{2 a}$

2. $\dfrac{\dfrac{3 b}{b-5}}{\dfrac{b^{2}}{b^{2}-25}}$

3. $\dfrac{\dfrac{5}{c^{2}+5 c-14}}{\dfrac{10}{c+7}}$

Answer: $\dfrac{1}{2(c-2)}$

4. $\dfrac{\dfrac{8}{d^{2}+9 d+18}}{\dfrac{12}{d+6}}$

5. $\dfrac{\dfrac{1}{2}+\dfrac{5}{6}}{\dfrac{2}{3}+\dfrac{7}{9}}$

Answer: $\dfrac{12}{13}$

6. $\dfrac{\dfrac{1}{2}+\dfrac{3}{4}}{\dfrac{3}{5}+\dfrac{7}{10}}$

7. $\dfrac{\dfrac{2}{3}-\dfrac{1}{9}}{\dfrac{3}{4}+\dfrac{5}{6}}$

Answer: $\dfrac{20}{57}$

8. $\dfrac{\dfrac{1}{2}-\dfrac{1}{6}}{\dfrac{2}{3}+\dfrac{3}{4}}$

9. $\dfrac{\dfrac{n}{m}+\dfrac{1}{n}}{\dfrac{1}{n}-\dfrac{n}{m}}$

Answer: $\dfrac{n^{2}+m}{m-n^{2}}$

10. $\dfrac{\dfrac{1}{p}+\dfrac{p}{q}}{\dfrac{q}{p}-\dfrac{1}{q}}$

11. $\dfrac{\dfrac{1}{r}+\dfrac{1}{t}}{\dfrac{1}{r^{2}}-\dfrac{1}{t^{2}}}$

Answer: $\dfrac{r t}{t-r}$

12. $\dfrac{\dfrac{2}{v}+\dfrac{2}{w}}{\dfrac{1}{v^{2}}-\dfrac{1}{w^{2}}}$

13. $\dfrac{x-\dfrac{2 x}{x+3}}{\dfrac{1}{x+3}+\dfrac{1}{x-3}}$

Answer: $\dfrac{(x+1)(x-3)}{2}$

14. $\dfrac{y-\dfrac{2 y}{y-4}}{\dfrac{2}{y-4}+\dfrac{2}{y+4}}$

15. $\dfrac{2-\dfrac{2}{a+3}}{\dfrac{1}{a+3}+\dfrac{a}{2}}$

Answer: $\dfrac{4}{a+1}$

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. $\dfrac{\dfrac{1}{3}+\dfrac{1}{8}}{\dfrac{1}{4}+\dfrac{1}{12}}$

Answer: $\dfrac{11}{8}$

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: $19$

20. $\dfrac{\dfrac{1}{6}+\dfrac{4}{15}}{\dfrac{3}{5}-\dfrac{1}{2}}$

21. $\dfrac{\dfrac{c}{d}+\dfrac{1}{d}}{\dfrac{1}{d}-\dfrac{d}{c}}$

Answer: $\dfrac{c^{2}+c}{c-d^{2}}$

22. $\dfrac{\dfrac{1}{m}+\dfrac{m}{n}}{\dfrac{n}{m}-\dfrac{1}{n}}$

23. $\dfrac{\dfrac{1}{p}+\dfrac{1}{q}}{\dfrac{1}{p^{2}}-\dfrac{1}{q^{2}}}$

Answer: $\dfrac{p q}{q-p}$

24. $\dfrac{\dfrac{2}{r}+\dfrac{2}{t}}{\dfrac{1}{r^{2}}-\dfrac{1}{t^{2}}}$

25. $\dfrac{\dfrac{2}{x+5}}{\dfrac{3}{x-5}+\dfrac{1}{x^{2}-25}}$

Answer: $\dfrac{2 x-10}{3 x+16}$

26. $\dfrac{\dfrac{5}{y-4}}{\dfrac{3}{y+4}+\dfrac{2}{y^{2}-16}}$

27. $\dfrac{\dfrac{5}{z^{2}-64}+\dfrac{3}{z+8}}{\dfrac{1}{z+8}+\dfrac{2}{z-8}}$

Answer: $\dfrac{3 z-19}{3 z+8}$

28. $\dfrac{\dfrac{3}{s+6}+\dfrac{5}{s-6}}{\dfrac{1}{s^{2}-36}+\dfrac{4}{s+6}}$

29. $\dfrac{\dfrac{4}{a^{2}-2 a-15}}{\dfrac{1}{a-5}+\dfrac{2}{a+3}}$

Answer: $\dfrac{4}{3 a-7}$

30. $\dfrac{\dfrac{5}{b^{2}-6 b-27}}{\dfrac{3}{b-9}+\dfrac{1}{b+3}}$

31. $\dfrac{\dfrac{5}{c+2}-\dfrac{3}{c+7}}{\dfrac{5 c}{c^{2}+9 c+14}}$

Answer: $\dfrac{2 c+29}{5 c}$

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: $\dfrac{2 p-5}{5}$

34. $\dfrac{\dfrac{n}{n-2}}{3+\dfrac{5}{n-2}}$

35. $\dfrac{\dfrac{m}{m+5}}{4+\dfrac{1}{m-5}}$

Answer: $\dfrac{m(m-5)}{(4 m-19)(m+5)}$

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: $\dfrac{13}{24}$

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: $2(a-4)$

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: $\dfrac{3 m n}{n-m}$

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: $\dfrac{(x-1)(x-2)}{6}$

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: 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.

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Simplify a Complex Rational Expression by Writing it as Division

Simplify a Complex Rational Expression by Using the LCD

Writing Exercises

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