Jay Abramson

Review Exercises

Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.

1.

${(5 - 3 \cdot 2)}^{2} - 6$

2.

$64 \div (2 \cdot 8) + 14 \div 7$

3.

$2 \cdot 5^{2} + 6 \div 2$

For the following exercises, solve the equation.

4.

$5 x + 9 = −11$

5.

$2 y + 4^{2} = 64$

For the following exercises, simplify the expression.

6.

$9 (y + 2) \div 3 \cdot 2 + 1$

7.

$3 m (4 + 7) - m$

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

8.

11

9.

0

10.

$\frac{5}{6}$

11.

$\sqrt{11}$

Exponents and Scientific Notation

For the following exercises, simplify the expression.

12.

$2^{2} \cdot 2^{4}$

13.

$\frac{4^{5}}{4^{3}}$

14.

${(\frac{a^{2}}{b^{3}})}^{4}$

15.

$\frac{6 a^{2} \cdot a^{0}}{2 a^{−4}}$

16.

$\frac{{(x y)}^{4}}{y^{3}} \cdot \frac{2}{x^{5}}$

17.

$\frac{4^{−2} x^{3} y^{−3}}{2 x^{0}}$

18.

${(\frac{2 x^{2}}{y})}^{−2}$

19.

$(\frac{16 a^{3}}{b^{2}}) {(4 a b^{−1})}^{−2}$

20.

Write the number in standard notation: $2.1314 \times 10^{−6}$

21.

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

Radicals and Rational Expressions

For the following exercises, find the principal square root.

22.

$\sqrt{121}$

23.

$\sqrt{196}$

24.

$\sqrt{361}$

25.

$\sqrt{75}$

26.

$\sqrt{162}$

27.

$\sqrt{\frac{32}{25}}$

28.

$\sqrt{\frac{80}{81}}$

29.

$\sqrt{\frac{49}{1250}}$

30.

$\frac{2}{4 + \sqrt{2}}$

31.

$4 \sqrt{3} + 6 \sqrt{3}$

32.

$12 \sqrt{5} - 13 \sqrt{5}$

33.

$\sqrt[5]{−243}$

34.

$\frac{\sqrt[3]{250}}{\sqrt[3]{−8}}$

Polynomials

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

35.

$(3 x^{3} + 2 x - 1) + (4 x^{2} - 2 x + 7)$

36.

$(2 y + 1) - (2 y^{2} - 2 y - 5)$

37.

$(2 x^{2} + 3 x - 6) + (3 x^{2} - 4 x + 9)$

38.

$(6 a^{2} + 3 a + 10) - (6 a^{2} −3 a + 5)$

39.

$(k + 3) (k - 6)$

40.

$(2 h + 1) (3 h - 2)$

41.

$(x + 1) (x^{2} + 1)$

42.

$(m - 2) (m^{2} + 2 m - 3)$

43.

$(a + 2 b) (3 a - b)$

44.

$(x + y) (x - y)$

Factoring Polynomials

For the following exercises, find the greatest common factor.

45.

$81 p + 9 p q - 27 p^{2} q^{2}$

46.

$12 x^{2} y + 4 x y^{2} −18 x y$

47.

$88 a^{3} b + 4 a^{2} b - 144 a^{2}$

For the following exercises, factor the polynomial.

48.

$2 x^{2} - 9 x - 18$

49.

$8 a^{2} + 30 a - 27$

50.

$d^{2} - 5 d - 66$

51.

$x^{2} + 10 x + 25$

52.

$y^{2} - 6 y + 9$

53.

$4 h^{2} - 12 h k + 9 k^{2}$

54.

$361 x^{2} - 121$

55.

$p^{3} + 216$

56.

$8 x^{3} - 125$

57.

$64 q^{3} - 27 p^{3}$

58.

$4 x {(x - 1)}^{- \frac{1}{4}} + 3 {(x - 1)}^{\frac{3}{4}}$

59.

$3 p {(p + 3)}^{\frac{1}{3}} −8 {(p + 3)}^{\frac{4}{3}}$

60.

$4 r {(2 r - 1)}^{- \frac{2}{3}} - 5 {(2 r - 1)}^{\frac{1}{3}}$

Rational Expressions

For the following exercises, simplify the expression.

61.

$\frac{x^{2} - x - 12}{x^{2} - 8 x + 16}$

62.

$\frac{4 y^{2} - 25}{4 y^{2} - 20 y + 25}$

63.

$\frac{2 a^{2} - a - 3}{2 a^{2} - 6 a - 8} \cdot \frac{5 a^{2} - 19 a - 4}{10 a^{2} - 13 a - 3}$

64.

$\frac{d - 4}{d^{2} - 9} \cdot \frac{d - 3}{d^{2} - 16}$

65.

$\frac{m^{2} + 5 m + 6}{2 m^{2} - 5 m - 3} \div \frac{2 m^{2} + 3 m - 9}{4 m^{2} - 4 m - 3}$

66.

$\frac{4 d^{2} - 7 d - 2}{6 d^{2} - 17 d + 10} \div \frac{8 d^{2} + 6 d + 1}{6 d^{2} + 7 d - 10}$

67.

$\frac{10}{x} + \frac{6}{y}$

68.

$\frac{12}{a^{2} + 2 a + 1} - \frac{3}{a^{2} −1}$

69.

$\frac{\frac{1}{d} + \frac{2}{c}}{\frac{6 c + 12 d}{d c}}$

70.

$\frac{\frac{3}{x} - \frac{7}{y}}{\frac{2}{x}}$