 Review Exercises

Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.

1 .

$( 5−3⋅2 ) 2 −6 ( 5−3⋅2 ) 2 −6$

2 .

$64÷( 2⋅8 )+14÷7 64÷( 2⋅8 )+14÷7$

3 .

$2⋅ 5 2 +6÷2 2⋅ 5 2 +6÷2$

For the following exercises, solve the equation.

4 .

$5x+9=−11 5x+9=−11$

5 .

$2y+ 4 2 =64 2y+ 4 2 =64$

For the following exercises, simplify the expression.

6 .

$9( y+2 )÷3⋅2+1 9( y+2 )÷3⋅2+1$

7 .

$3m( 4+7 )−m 3m( 4+7 )−m$

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

8 .

11

9 .

0

10 .

$5 6 5 6$

11 .

$11 11$

Exponents and Scientific Notation

For the following exercises, simplify the expression.

12 .

$2 2 ⋅ 2 4 2 2 ⋅ 2 4$

13 .

$4 5 4 3 4 5 4 3$

14 .

$( a 2 b 3 ) 4 ( a 2 b 3 ) 4$

15 .

$6 a 2 ⋅ a 0 2 a −4 6 a 2 ⋅ a 0 2 a −4$

16 .

$( xy ) 4 y 3 ⋅ 2 x 5 ( xy ) 4 y 3 ⋅ 2 x 5$

17 .

$4 −2 x 3 y −3 2 x 0 4 −2 x 3 y −3 2 x 0$

18 .

$( 2 x 2 y ) −2 ( 2 x 2 y ) −2$

19 .

$( 16 a 3 b 2 ) ( 4a b −1 ) −2 ( 16 a 3 b 2 ) ( 4a b −1 ) −2$

20 .

Write the number in standard notation: $2.1314× 10 −6 2.1314× 10 −6$

21 .

Write the number in scientific notation: 16,340,000

For the following exercises, find the principal square root.

22 .

$121 121$

23 .

$196 196$

24 .

$361 361$

25 .

$75 75$

26 .

$162 162$

27 .

$32 25 32 25$

28 .

$80 81 80 81$

29 .

$49 1250 49 1250$

30 .

$2 4+ 2 2 4+ 2$

31 .

$4 3 +6 3 4 3 +6 3$

32 .

$12 5 −13 5 12 5 −13 5$

33 .

$−243 5 −243 5$

34 .

$250 3 −8 3 250 3 −8 3$

Polynomials

For the following exercises, perform the given operations and simplify.

35 .

$(3 x 3 +2x−1)+(4 x 2 −2x+7) (3 x 3 +2x−1)+(4 x 2 −2x+7)$

36 .

$( 2y+1 )−( 2 y 2 −2y−5 ) ( 2y+1 )−( 2 y 2 −2y−5 )$

37 .

$(2 x 2 +3x−6)+(3 x 2 −4x+9) (2 x 2 +3x−6)+(3 x 2 −4x+9)$

38 .

$( 6 a 2 +3a+10 )−( 6 a 2 −3a+5 ) ( 6 a 2 +3a+10 )−( 6 a 2 −3a+5 )$

39 .

$(k+3)(k−6) (k+3)(k−6)$

40 .

$(2h+1)(3h−2) (2h+1)(3h−2)$

41 .

$( x+1 )( x 2 +1 ) ( x+1 )( x 2 +1 )$

42 .

$(m−2)( m 2 +2m−3) (m−2)( m 2 +2m−3)$

43 .

$( a+2b )( 3a−b ) ( a+2b )( 3a−b )$

44 .

$( x+y )( x−y ) ( x+y )( x−y )$

Factoring Polynomials

For the following exercises, find the greatest common factor.

45 .

$81p+9pq−27 p 2 q 2 81p+9pq−27 p 2 q 2$

46 .

$12 x 2 y+4x y 2 −18xy 12 x 2 y+4x y 2 −18xy$

47 .

$88 a 3 b+4 a 2 b−144 a 2 88 a 3 b+4 a 2 b−144 a 2$

For the following exercises, factor the polynomial.

48 .

$2 x 2 −9x−18 2 x 2 −9x−18$

49 .

$8 a 2 +30a−27 8 a 2 +30a−27$

50 .

$d 2 −5d−66 d 2 −5d−66$

51 .

$x 2 +10x+25 x 2 +10x+25$

52 .

$y 2 −6y+9 y 2 −6y+9$

53 .

$4 h 2 −12hk+9 k 2 4 h 2 −12hk+9 k 2$

54 .

$361 x 2 −121 361 x 2 −121$

55 .

$p 3 +216 p 3 +216$

56 .

$8 x 3 −125 8 x 3 −125$

57 .

$64 q 3 −27 p 3 64 q 3 −27 p 3$

58 .

$4x (x−1) − 1 4 +3 (x−1) 3 4 4x (x−1) − 1 4 +3 (x−1) 3 4$

59 .

$3p ( p+3 ) 1 3 −8 ( p+3 ) 4 3 3p ( p+3 ) 1 3 −8 ( p+3 ) 4 3$

60 .

$4r ( 2r−1 ) − 2 3 −5 ( 2r−1 ) 1 3 4r ( 2r−1 ) − 2 3 −5 ( 2r−1 ) 1 3$

Rational Expressions

For the following exercises, simplify the expression.

61 .

$x 2 −x−12 x 2 −8x+16 x 2 −x−12 x 2 −8x+16$

62 .

$4 y 2 −25 4 y 2 −20y+25 4 y 2 −25 4 y 2 −20y+25$

63 .

$2 a 2 −a−3 2 a 2 −6a−8 ⋅ 5 a 2 −19a−4 10 a 2 −13a−3 2 a 2 −a−3 2 a 2 −6a−8 ⋅ 5 a 2 −19a−4 10 a 2 −13a−3$

64 .

$d−4 d 2 −9 ⋅ d−3 d 2 −16 d−4 d 2 −9 ⋅ d−3 d 2 −16$

65 .

$m 2 +5m+6 2 m 2 −5m−3 ÷ 2 m 2 +3m−9 4 m 2 −4m−3 m 2 +5m+6 2 m 2 −5m−3 ÷ 2 m 2 +3m−9 4 m 2 −4m−3$

66 .

$4 d 2 −7d−2 6 d 2 −17d+10 ÷ 8 d 2 +6d+1 6 d 2 +7d−10 4 d 2 −7d−2 6 d 2 −17d+10 ÷ 8 d 2 +6d+1 6 d 2 +7d−10$

67 .

$10 x + 6 y 10 x + 6 y$

68 .

$12 a 2 +2a+1 − 3 a 2 −1 12 a 2 +2a+1 − 3 a 2 −1$

69 .

$1 d + 2 c 6c+12d dc 1 d + 2 c 6c+12d dc$

70 .

$3 x − 7 y 2 x 3 x − 7 y 2 x$