Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Simplify each rational expression. State any restrictions on the variables.

8. − 5 x 3 y 15 x y 3 negative . fraction 5 , x cubed , y , over 15 x , y cubed end fraction
9. 2 x 4 x 2 − 2 x fraction 2 x , over 4 , x squared , minus 2 x end fraction
10. 6 c 2 + 9 c 3 c fraction 6 , c squared , plus 9 c , over 3 c end fraction
11. 49 − z 2 z + 7 fraction 49 minus , z squared , over z plus 7 end fraction
12. x 2 + 8 x + 16 x 2 − 2 x − 24 fraction x squared , plus 8 x plus 16 , over x squared , minus 2 x minus 24 end fraction
13. 12 − x − x 2 x 2 − 8 x + 15 fraction 12 minus x minus , x squared , over x squared , minus 8 x plus 15 end fraction

See Problem 2.

Multiply. State any restrictions on the variables.

14. 4 x 2 5 y ⋅ 7 y 12 x 4 fraction 4 , x squared , over 5 y end fraction . dot . fraction 7 y , over 12 , x to the fourth end fraction
15. 2 x 4 10 y − 2 ⋅ 5 y 3 4 x 3 fraction 2 , x to the fourth , over 10 , y super negative 2 end super end fraction . dot . fraction 5 , y cubed , over 4 , x cubed end fraction
16. 8 y − 4 10 y − 5 ⋅ 5 y − 15 3 y − 9 fraction 8 y minus 4 , over 10 y minus 5 end fraction . dot . fraction 5 y minus 15 , over 3 y minus 9 end fraction
17. 2 x + 12 3 x − 9 ⋅ 6 − 2 x 3 x + 8 fraction 2 x plus 12 , over 3 x minus 9 end fraction . dot . fraction 6 minus 2 x , over 3 x plus 8 end fraction
18. x 2 − 4 x 2 − 1 ⋅ x + 1 x 2 + 2 x fraction x squared , minus 4 , over x squared , minus 1 end fraction . dot . fraction x plus 1 , over x squared , plus 2 x end fraction
19. x 2 − 5 x + 6 x 2 − 4 ⋅ x 2 + 3 x + 2 x 2 − 2 x − 3 fraction x squared , minus 5 x plus 6 , over x squared , minus 4 end fraction . dot . fraction x squared , plus 3 x plus 2 , over x squared , minus 2 x minus 3 end fraction

See Problem 3.

Divide. State any restrictions on the variables.

20. 7 x 4 y 3 ÷ 21 x 3 8 y fraction 7 x , over 4 , y cubed end fraction . divides . fraction 21 , x cubed , over 8 y end fraction
21. 3 x 3 5 y 2 ÷ 6 y − 3 5 x − 5 fraction 3 , x cubed , over 5 , y squared end fraction . divides . fraction 6 , y super negative 3 end super , over 5 , x super negative 5 end super end fraction
22. 6 x + 6 y y − x ÷ 18 5 x − 5 y fraction 6 x plus 6 y , over y minus x end fraction . divides . fraction 18 , over 5 x minus 5 y end fraction
23. 3 y − 12 2 y + 4 ÷ 6 y − 24 8 + 4 y fraction 3 y minus 12 , over 2 y plus 4 end fraction . divides . fraction 6 y minus 24 , over 8 plus 4 y end fraction
24. x 2 x 2 + 2 x + 1 ÷ 3 x x 2 − 1 fraction x squared , over x squared , plus 2 x plus 1 end fraction . divides . fraction 3 x , over x squared , minus 1 end fraction
25. y 2 − 5 y + 6 y 3 ÷ y 2 + 3 y − 10 4 y 2 fraction y squared , minus 5 y plus 6 , over y cubed end fraction . divides . fraction y squared , plus 3 y minus 10 , over 4 , y squared end fraction

    See Problem 4.

26. Industrial Design A storage tank will have a circular base of radius r and a height of r. The tank can be either cylindrical or hemispherical (half a sphere).

    1. Write and simplify an expression for the ratio of the volume of the hemispherical tank to its surface area (including the base). For a sphere, V = 4 3 π r 3 v equals , 4 thirds , pi , r cubed  and SA = 4 π r 2 .
    2. Write and simplify an expression for the ratio of the volume of the cylindrical tank to its surface area (including the bases).
    3. Compare the ratios of volume to surface area for the two tanks.
    4. Compare the volumes of the two tanks.

B Apply

Simplify each rational expression. State any restrictions on the variables.

27. x 2 − 5 x − 24 x 2 − 7 x − 30 fraction x squared , minus 5 x minus 24 , over x squared , minus 7 x minus 30 end fraction
28. 2 y 2 + 8 y − 24 2 y 2 − 8 y + 8 fraction 2 , y squared , plus 8 y minus 24 , over 2 , y squared , minus 8 y plus 8 end fraction
29. x y 3 − 9 x y 12 x y 2 + 12 x y − 144 x fraction x , y cubed , minus 9 x y , over 12 x , y squared , plus 12 x y minus 144 x end fraction
30. Open-Ended Write three rational expressions that simplify to x x + 1 . fraction x , over x plus 1 end fraction . .

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