31. Think About a Plan A cereal company wants to use the most efficient packaging for their new product. They are considering a cylindrical-shaped box and a cube-shaped box. Compare the ratios of the volume to the surface area of the containers to determine which packaging will be more efficient.
    • How can you measure the cereal box's efficiency?
    • What formulas will you need to use to solve this problem?

Multiply or divide. State any restrictions on the variables.

32. 6 x 3 − 6 x 2 x 4 + 5 x 3 ÷ 3 x 2 − 15 x + 12 2 x 2 + 2 x − 40 fraction 6 , x cubed , minus 6 , x squared , over x to the fourth , plus 5 , x cubed end fraction . divides . fraction 3 , x squared , minus 15 x plus 12 , over 2 , x squared , plus 2 x minus 40 end fraction
33. 2 x 2 − 6 x x 2 + 18 x + 81 ⋅ 9 x + 81 x 2 − 9 fraction 2 , x squared , minus 6 x , over x squared , plus 18 x plus 81 end fraction . dot . fraction 9 x plus 81 , over x squared , minus 9 end fraction
34. x 2 − x − 2 2 x 2 − 5 x + 2 ÷ x 2 − x − 12 2 x 2 + 5 x − 3 fraction x squared , minus x minus 2 , over 2 , x squared , minus 5 x plus 2 end fraction . divides . fraction x squared , minus x minus 12 , over 2 , x squared , plus 5 x minus 3 end fraction
35. 2 x 2 + 5 x + 2 4 x 2 − 1 ⋅ 2 x 2 + x − 1 x 2 + x − 2 fraction 2 , x squared , plus 5 x plus 2 , over 4 , x squared , minus 1 end fraction . dot . fraction 2 , x squared , plus x minus 1 , over x squared , plus x minus 2 end fraction

36. Reasoning Write a simplified expression for the area of the rectangle below. State all restrictions on a.

    

37. Manufacturing A toy company is considering a cube or sphere-shaped container for packaging a new product. The height of the cube would equal the diameter of the sphere. Compare the volume-to-surface area ratios of the containers. Which packaging will be more efficient? For a sphere, S A = 4 π r 2 . s eh equals 4 pi , r squared , .

Decide whether the given statement is always, sometimes, or never true.

38. Rational expressions contain exponents.
39. Rational expressions contain logarithms.
40. Rational expressions are undefined for values of the variables that make the denominator 0.
41. Restrictions on variables change when a rational expression is simplified.

Simplify. State any restrictions on the variables.

42. ( x 2 − x ) 2 x ( x − 1 ) − 2 ( x 2 + 3 x − 4 ) fraction open . x squared , minus x . close squared , over x . open , x minus 1 , close super negative 2 end super . open . x squared , plus 3 x minus 4 . close end fraction
43. 2 x + 6 ( x − 1 ) − 1 ( x 2 + 2 x − 3 ) fraction 2 x plus 6 , over open , x minus 1 , close super negative 1 end super . open . x squared , plus 2 x minus 3 . close end fraction
44. 54 x 3 y − 1 3 x − 2 y fraction 54 , x cubed . y super negative 1 end super , over 3 , x super negative 2 end super , y end fraction

C Challenge

45. 1. Reasoning Simplify ( 2 x n ) 2 − 1 2 x n − 1 , fraction open , 2 , x to the n , close squared . minus 1 , over 2 , x to the n , minus 1 end fraction . comma  where x is an integer and n is a positive integer. (Hint: Factor the numerator.)
    2. Use the result from part (a) to show that the value of the given expression is always an odd integer.

Use the fact that a b c d = a b ÷ c d fraction fraction eh , over b end fraction , over fraction c , over d end fraction end fraction . equals , eh over b , divides , c over d  to simplify each rational expression. State any restrictions on the variables.

46. 8 x 2 y x + 1 6 x y 2 x + 1 fraction fraction 8 , x squared , y , over x plus 1 end fraction , over fraction 6 x , y squared , over x plus 1 end fraction end fraction
47. 3 a 3 b 3 a − b 4 a b b − a fraction fraction 3 , eh cubed , b cubed , over eh minus b end fraction , over fraction 4 eh b , over b minus eh end fraction end fraction
48. 9 m + 6 n m 2 n 2 12 m + 8 n 5 m 2 fraction fraction 9 m plus 6 n , over m squared , n squared end fraction , over fraction 12 m plus 8 n , over 5 , m squared end fraction end fraction
49. x 2 − 1 x 2 − 9 x 2 + 3 x − 4 x 2 + 8 x + 15 fraction fraction x squared , minus 1 , over x squared , minus 9 end fraction , over fraction x squared , plus 3 x minus 4 , over x squared , plus 8 x plus 15 end fraction end fraction

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