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# 7.4E: 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. $$\dfrac{\dfrac{2 a}{a+4}}{\dfrac{4 a^{2}}{a^{2}-16}}$$

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

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

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

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

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

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

$$\dfrac{12}{13}$$

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

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

$$\dfrac{20}{57}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\dfrac{11}{8}$$

18. $$\dfrac{\dfrac{1}{4}+\dfrac{1}{9}}{\dfrac{1}{6}+\dfrac{1}{12}}$$

19. $$\dfrac{\dfrac{5}{6}+\dfrac{2}{9}}{\dfrac{7}{18}-\dfrac{1}{3}}$$

$$19$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

32. $$\dfrac{\dfrac{6}{d-4}-\dfrac{2}{d+7}}{\dfrac{2 d}{d^{2}+3 d-28}}$$

33. $$\dfrac{2+\dfrac{1}{p-3}}{\dfrac{5}{p-3}}$$

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

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

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

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

36. $$\dfrac{7+\dfrac{2}{q-2}}{\dfrac{1}{q+2}}$$

In the following exercises, simplify each complex rational expression using either method.

37. $$\dfrac{\dfrac{3}{4}-\dfrac{2}{7}}{\dfrac{1}{2}+\dfrac{5}{14}}$$

$$\dfrac{13}{24}$$

38. $$\dfrac{\dfrac{v}{w}+\dfrac{1}{v}}{\dfrac{1}{v}-\dfrac{v}{w}}$$

39. $$\dfrac{\dfrac{2}{a+4}}{\dfrac{1}{a^{2}-16}}$$

$$2(a-4)$$

40. $$\dfrac{\dfrac{3}{b^{2}-3 b-40}}{\dfrac{5}{b+5}-\dfrac{2}{b-8}}$$

41. $$\dfrac{\dfrac{3}{m}+\dfrac{3}{n}}{\dfrac{1}{m^{2}}-\dfrac{1}{n^{2}}}$$

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

42. $$\dfrac{\dfrac{2}{r-9}}{\dfrac{1}{r+9}+\dfrac{3}{r^{2}-81}}$$

43. $$\dfrac{x-\dfrac{3 x}{x+2}}{\dfrac{3}{x+2}+\dfrac{3}{x-2}}$$

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

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?

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.