第1章前提知識Prerequisites

Review Exercises復習問題

Real Numbers: Algebra Essentials実数:代数の基礎

For the following exercises, perform the given operations.

次の各問について、与えられた計算を行え。

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.

次の各問について、方程式を解け。

4. $5 x + 9 = −11$

5. $2 y + 4^{2} = 64$

For the following exercises, simplify the expression.

次の各問について、式を簡単にせよ。

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.

次の各問について、その数が有理数・無理数・非負整数・自然数のどれにあたるかを答えよ。もっともよく言い当てているものを選べ。

8. 11

9. 0

10. $\frac{5}{6}$

11. $\sqrt{11}$

Exponents and Scientific Notation指数と科学的記数法

For the following exercises, simplify the expression.

次の各問について、式を簡単にせよ。

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: $2.1314 \times 10^{−6}$

20. 次の数を通常の表記で書け。 $2.1314 \times 10^{−6}$

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

21. 次の数を科学的記数法で書け。16,340,000

Radicals and Rational Expressions根号と有理式

For the following exercises, find the principal square root.

次の各問について、正の平方根を求めよ。

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多項式

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

次の各問について、与えられた計算を行い、簡単にせよ。

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多項式の因数分解

For the following exercises, find the greatest common factor.

次の各問について、最大公約因数を求めよ。

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.

次の各問について、多項式を因数分解せよ。

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有理式

For the following exercises, simplify the expression.

次の各問について、式を簡単にせよ。

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