1.E: The Arithmetic of Numbers (Exercises)

1.1 An Introduction to the Integers

In Exercises 1-8, simplify each of the following expressions.

1) $|5|$

Answer

$5$

2) $|1|$

3) $|-2|$

Answer

$2$

4) $|-1|$

5) $|2|$

Answer

$2$

6) $|8|$

7) $|-4|$

Answer

$4$

8) $|-6|$

In Exercises 9-24, simplify each of the following expressions as much as possible.

9) $-91+(-147)$

Answer

$-238$

10) $-23+(-13)$

11) $96+145$

Answer

$241$

12) $16+127$

13) $-76+46$

Answer

$-30$

14) $-11+21$

15) $-59+(-12)$

Answer

$-71$

16) $-40+(-58)$

17) $37+(-86)$

Answer

$-49$

18) $143+(-88)$

19) $66+(-85)$

Answer

$-19$

20) $33+(-41)$

21) $57+20$

Answer

$77$

22) $66+110$

23) $-48+127$

Answer

$79$

24) $-48+92$

In Exercises 25-32, find the difference.

25) $-20-(-10)$

Answer

$-10$

26) $-20-(-20)$

27) $-62-7$

Answer

$-69$

28) $-82-62$

29) $-77-26$

Answer

$-103$

30) $-96-92$

31) $-7-(-16)$

Answer

$9$

32) $-20-(-5)$

In Exercises 33-40, compute the exact value.

33) $(-8)^{6}$

Answer

$262144$

34) $(-3)^{5}$

35) $(-7)^{5}$

Answer

$−16807$

36) $(-4)^{6}$

37) $(-9)^{2}$

Answer

$81$

38) $(-4)^{2}$

39) $(-4)^{4}$

Answer

$256$

40) $(-5)^{4}$

In Exercises 41-52, use your graphing calculator to compute the given expression.

41) $-562-1728$

Answer

$−2290$

42) $-3125-(-576)$

43) $-400-(-8225)$

Answer

$7825$

44) $-8176+578$

45) $(-856)(232)$

Answer

$−198592$

46) $(-335)(-87)$

47) $(-815)(-3579)$

Answer

$2916885$

48) $(753)(-9753)$

49) $(-18)^{3}$

Answer

$−5832$

50) $(-16)^{4}$

51) $(-13)^{5}$

Answer

$−371293$

52) $(-15)^{6}$

1.2 Order of Operations

In Exercises 1-18, simplify the given expression.

1) $-12+6(-4)$

Answer

$-36$

2) $11+11(7)$

3) $-(-2)^{5}$

Answer

$32$

4) $-(-5)^{3}$

5) $-|-40|$

Answer

$-40$

6) $-|-42|$

7) $-24 /(-6)(-1)$

Answer

$-4$

8) 45$/(-3)(3)$

9) $-(-50)$

Answer

$50$

10) $-(-30)$

11) $-3^{5}$

Answer

$-243$

12) $-3^{2}$

13) $48 \div 4(6)$

Answer

$72$

14) $96 \div 6(4)$

15) $-52-8(-8)$

Answer

$12$

16) $-8-7(-3)$

17) $(-2)^{4}$

Answer

$16$

18) $(-4)^{4}$

In Exercises 19-42, simplify the given expression.

19) $9-3(2)^{2}$

Answer

$-3$

20) $-4-4(2)^{2}$

21) $17-10|13-14|$

Answer

$7$

22) $18-3|-20-5|$

23) $-4+5(-4)^{3}$

Answer

$−324$

24) $3+3(-4)^{3}$

25) $8+5(-1-6)$

Answer

$-27$

26) $8+4(-5-5)$

27) $(10-8)^{2}-(7-5)^{3}$

Answer

$-4$

28) $(8-10)^{2}-(4-5)^{3}$

29) $6-9(6-4(9-7))$

Answer

$24$

30) $4-3(3-5(7-2))$

31) $-6-5(4-6)$

Answer

$4$

32) $-5-5(-7-7)$

33) $9+(9-6)^{3}-5$

Answer

$31$

34) $12+(8-3)^{3}-6$

35) $-5+3(4)^{2}$

Answer

$43$

36) $2+3(2)^{2}$

37) $8-(5-2)^{3}+6$

Answer

$-13$

38) $9-(12-11)^{2}+4$

39) $|6-15|-|-17-11|$

Answer

$-19$

40) $|-18-19|-|-3-12|$

41) $5-5(5-6(6-4))$

Answer

$40$

42) $4-6(4-7(8-5))$

In Exercises 43-58, evaluate the expression at the given values of $x$ and $y$.

43) $4 x^{2}+3 x y+4 y^{2}$ at $x=-3$ and $y=0$

Answer

$36$

44) $3 x^{2}-3 x y+2 y^{2}$ at $x=4$ and $y=-3$

45) $-8 x+9$ at $x=-9$

Answer

$81$

46) $-12 x+10$ at $x=2$

47) $-5 x^{2}+2 x y-4 y^{2}$ at $x=5$ and $y=0$

Answer

$-125$

48) $3 x^{2}+3 x y-5 y^{2}$ at $x=0$ and $y=3$

49) $3 x^{2}+3 x-4$ at $x=5$

Answer

$86$

50) $2 x^{2}+6 x-5$ at $x=6$

51) $-2 x^{2}+2 y^{2}$ at $x=1$ and $y=-2$

Answer

$6$

52) $-5 x^{2}+5 y^{2}$ at $x=-4$ and $y=0$

53) $-3 x^{2}-6 x+3$ at $x=2$

Answer

$−21$

54) $-7 x^{2}+9 x+5$ at $x=-7$

55) $-6 x-1$ at $x=1$

Answer

$−7$

56) $10 x+7$ at $x=9$

57) $3 x^{2}-2 y^{2}$ at $x=-3$ and $y=-2$

Answer

$19$

58) $-3 x^{2}+2 y^{2}$ at $x=2$ and $y=2$

59) Evaluate $\dfrac{a^{2}+b^{2}}{a+b}$ at $a = 27$ and $b =−30$.

Answer

$-543$

60) Evaluate $\dfrac{a^{2}+b^{2}}{a+b}$ at $a = −63$ and $b = 77$.

61) Evaluate $\dfrac{a+b}{c-d}$ at $a = −42$, $b = 25$, $c = 26$, and $d = 43$.

Answer

$1$

62) Evaluate $\dfrac{a+b}{c-d}$ at $a = 38$, $b = 42$, $c = 10$, and $d = 50$.

63) Evaluate $\dfrac{a-b}{c d}$ at $a =−7$, $b = 48$, $c = 5$, and $d = 11$.

Answer

$-1$

64) Evaluate $\dfrac{a-b}{c d}$ at $a =−46$, $b = 46$, $c = 23$, and $d = 2$.

65) Evaluate the expressions $a^{2}+b^{2}$ and $(a+b)^{2}$ at $a = 3$ and $b = 4$. Do the expressions produce the same results?

Answer

No

66) Evaluate the expressions $a^{2} b^{2}$ and $(ab)^2$ at $a = 3$ and $b = 4$. Do the expressions produce the same results?

67) Evaluate the expressions $|a||b|$ and $|ab|$ at $a = −3$ and $b = 5$. Do the expressions produce the same results?

Answer

Yes

68) Evaluate the expressions $|a|+|b|$ and $|a + b|$ at $a = −3$ and $b = 5$. Do the expressions produce the same results?

In Exercises 69-72, use a graphing calculator to evaluate the given expression.

69) $-236-324(-576+57)$

Answer

$167920$

70) $-443+27(-414-22)$

71) $\dfrac{270-900}{300-174}$

Answer

$-5$

72) $\dfrac{3000-952}{144-400}$

73) Use a graphing calculator to evaluate the expression $\dfrac{a^{2}+b^{2}}{a+b}$ at $a = −93$ and $b = 84$ by first storing $−93$ in the variable $A$ and $84$ in the variable $B$, then entering the expression $(A^2+B^2)/(A+B)$.

Answer

$−1745$

74) Use a graphing calculator to evaluate the expression $\dfrac{a^{2}+b^{2}}{a+b}$ at $a = −76$ and $b = 77$ by first storing $−76$ in the variable $A$ and $77$ in the variable $B$, then entering the expression $(A^2+B^2)/(A+B)$.

75) The formula $F=\dfrac{9}{5} C+32$ will change a Celsius temperature to a Fahrenheit temperature. Given that the Celsius temperature is $C=60^{\circ} \mathrm{C}$, find the equivalent Fahrenheit temperature.

Answer

$140^{\circ} \mathrm{F}$

76) The surface area of a cardboard box is given by the formula $$S =2WH+2LH +2LW \nonumber$$ where $W$ and $L$ are the width and length of the base of the box and $H$ is its height. If $W = 2$ centimeters, $L = 8$ centimeters, and $H = 2$ centimeters, find the surface area of the box.

77) The kinetic energy (in joules) of an object having mass $m$ (in kilograms) and velocity $v$ (in meters per second) is given by the formula $K=\dfrac{1}{2} m v^{2}$. Given that the mass of the object is $m =7$ kilograms and its velocity is $v = 50$ meters per second, calculate the kinetic energy of the object.

Answer

$8750$ joules

78) The area of a trapezoid is given by the formula $A=\dfrac{1}{2}\left(b_{1}+b_{2}\right) h$, where $b_1$ and $b_2$ are the lengths of the parallel bases and $h$ is the height of the trapezoid. If the lengths of the bases are $21$ yards and $11$ yards, respectively, and if the height is $22$ yards, find the area of the trapezoid.

1.3 The Rational Numbers

In Exercises 1-6, reduce the given fraction to lowest terms by dividing numerator and denominator by the their greatest common divisor.

1) $\dfrac{20}{50}$

Answer

$\dfrac{2}{5}$

2) $\dfrac{36}{38}$

3) $\dfrac{10}{48}$

Answer

$\dfrac{5}{24}$

4) $\dfrac{36}{14}$

5) $\dfrac{24}{45}$

Answer

$\dfrac{8}{15}$

6) $\dfrac{21}{36}$

In Exercises 7-12, reduce the given fraction to lowest terms by prime factoring both numerator and denominator and canceling common factors.

7) $\dfrac{153}{170}$

Answer:

$\dfrac{9}{10}$

8) $\dfrac{198}{144}$

In Exercises 9-24, simplify each of the following expressions as much as possible.

9) $\dfrac{188}{141}$

Answer

$\dfrac{4}{3}$

10) $\dfrac{171}{144}$

11) $\dfrac{159}{106}$

Answer

$\dfrac{3}{2}$

12) $\dfrac{140}{133}$

In Exercises 13-18, for each of the following problems, multiply numerators and denominators, then prime factor and cancel to reduce your answer to lowest terms.

13) $\dfrac{20}{8} \cdot\left(-\dfrac{18}{13}\right)$

Answer

$-\dfrac{45}{13}$

14) $\dfrac{18}{16} \cdot\left(-\dfrac{2}{5}\right)$

15) $-\dfrac{19}{4} \cdot\left(-\dfrac{18}{13}\right)$

Answer

$\dfrac{171}{26}$

16) $-\dfrac{3}{2} \cdot\left(-\dfrac{14}{6}\right)$

17) $-\dfrac{16}{8} \cdot \dfrac{19}{6}$

Answer

$-\dfrac{19}{3}$

18) $-\dfrac{14}{4} \cdot \dfrac{7}{17}$

In Exercises 19-24, for each of the following problems, first prime factor all numerators and denominators, then cancel. After canceling, multiply numerators and denominators.

19) $-\dfrac{5}{6} \cdot\left(-\dfrac{12}{49}\right)$

Answer

$\dfrac{10}{49}$

20) $-\dfrac{36}{17} \cdot\left(-\dfrac{21}{46}\right)$

21) $-\dfrac{21}{10} \cdot \dfrac{12}{55}$

Answer

$-\dfrac{126}{275}$

22) $-\dfrac{49}{13} \cdot \dfrac{52}{51}$

23) $\dfrac{55}{29} \cdot\left(-\dfrac{54}{11}\right)$

Answer

$-\dfrac{270}{29}$

24) $\dfrac{7}{13} \cdot\left(-\dfrac{55}{49}\right)$

In Exercises 25-30, divide. Be sure your answer is reduced to lowest terms.

25) $\dfrac{50}{39} \div\left(-\dfrac{5}{58}\right)$

Answer

$-\dfrac{580}{39}$

26) $\dfrac{31}{25} \div\left(-\dfrac{4}{5}\right)$

27) $-\dfrac{60}{17} \div \dfrac{34}{31}$

Answer

$-\dfrac{930}{289}$

28) $-\dfrac{27}{28} \div \dfrac{45}{23}$

29) $-\dfrac{7}{10} \div\left(-\dfrac{13}{28}\right)$

Answer

$\dfrac{98}{65}$

30) $-\dfrac{4}{13} \div\left(-\dfrac{48}{35}\right)$

In Exercises 31-38, add or subtract the fractions, as indicated, and simplify your result.

31) $-\dfrac{5}{6}+\dfrac{1}{4}$

Answer

$-\dfrac{7}{12}$

32) $-\dfrac{1}{7}+\dfrac{5}{8}$

33) $-\dfrac{8}{9}+\left(-\dfrac{1}{3}\right)$

Answer

$-\dfrac{11}{9}$

34) $-\dfrac{1}{3}+\left(-\dfrac{1}{2}\right)$

35) $-\dfrac{1}{4}-\left(-\dfrac{2}{9}\right)$

Answer

$-\dfrac{1}{36}$

36) $-\dfrac{1}{2}-\left(-\dfrac{1}{8}\right)$

37) $-\dfrac{8}{9}-\dfrac{4}{5}$

Answer

$-\dfrac{76}{45}$

38) $-\dfrac{4}{7}-\dfrac{1}{3}$

In Exercises 39-52, simplify the expression.

39) $\dfrac{8}{9}-\left|\dfrac{5}{2}-\dfrac{2}{5}\right|$

Answer

$-\dfrac{109}{90}$

40) $\dfrac{8}{5}-\left|\dfrac{7}{6}-\dfrac{1}{2}\right|$

41) $\left(-\dfrac{7}{6}\right)^{2}+\left(-\dfrac{1}{2}\right)\left(-\dfrac{5}{3}\right)$

Answer

$\dfrac{79}{36}$

42) $\left(\dfrac{3}{2}\right)^{2}+\left(-\dfrac{1}{2}\right)\left(\dfrac{5}{8}\right)$

43) $\left(-\dfrac{9}{5}\right)\left(-\dfrac{9}{7}\right)+\left(\dfrac{8}{5}\right)\left(-\dfrac{1}{2}\right)$

Answer

$\dfrac{53}{35}$

44) $\left(-\dfrac{1}{3}\right)\left(-\dfrac{5}{7}\right)+\left(\dfrac{2}{3}\right)\left(-\dfrac{6}{7}\right)$

45) $-\dfrac{5}{8}+\dfrac{7}{2}\left(-\dfrac{9}{2}\right)$

Answer

$-\dfrac{131}{8}$

46) $\dfrac{3}{2}+\dfrac{9}{2}\left(-\dfrac{1}{4}\right)$

47) $\left(-\dfrac{7}{5}\right)\left(\dfrac{9}{2}\right)-\left(-\dfrac{2}{5}\right)^{2}$

Answer

$-\dfrac{323}{50}$

48) $\left(\dfrac{3}{4}\right)\left(\dfrac{2}{3}\right)-\left(\dfrac{1}{4}\right)^{2}$

49) $\dfrac{6}{5}-\dfrac{2}{5}\left(-\dfrac{4}{9}\right)$

Answer

$\dfrac{62}{45}$

50) $\dfrac{3}{2}-\dfrac{5}{6}\left(-\dfrac{1}{3}\right)$

51) $\left(\dfrac{2}{3}\right)\left(-\dfrac{8}{7}\right)-\left(\dfrac{4}{7}\right)\left(-\dfrac{9}{8}\right)$

Answer

$-\dfrac{5}{42}$

52) $\left(-\dfrac{3}{2}\right)\left(\dfrac{1}{3}\right)-\left(\dfrac{5}{8}\right)\left(-\dfrac{1}{8}\right)$

In Exercises 53-70, evaluate the expression at the given values.

53) $x y-z^{2}$ at $x=-1 / 2, y=-1 / 3,$ and $z=5 / 2$

Answer

$-\dfrac{73}{12}$

54) $x y-z^{2}$ at $x=-1 / 3, y=5 / 6,$ and $z=1 / 3$

55) $-5 x^{2}+2 y^{2}$ at $x=3 / 4$ and $y=-1 / 2$

Answer

$-\dfrac{37}{16}$

56) $-2 x^{2}+4 y^{2}$ at $x=4 / 3$ and $y=-3 / 2$

57) $2 x^{2}-2 x y-3 y^{2}$ at $x=3 / 2$ and $y=-3 / 4$

Answer

$\dfrac{81}{16}$

58) $5 x^{2}-4 x y-3 y^{2}$ at $x=1 / 5$ and $y=-4 / 3$

59) $x+y z$ at $x=-1 / 3, y=1 / 6,$ and $z=2 / 5$

Answer

$-\dfrac{2}{5}$

60) $x+y z$ at $x=1 / 2, y=7 / 4,$ and $z=2 / 3$

61) $a b+b c$ at $a=-4 / 7, b=7 / 5,$ and $c=-5 / 2$

Answer

$-\dfrac{43}{10}$

62) $a b+b c$ at $a=-8 / 5, b=7 / 2,$ and $c=-9 / 7$

63) $x^{3}$ at $x=-1 / 2$

Answer

$-\dfrac{1}{8}$

64) $x^{2}$ at $x=-3 / 2$

65) $x-y z$ at $x=-8 / 5, y=1 / 3,$ and $z=-8 / 5$

Answer

$-\dfrac{16}{15}$

66) $x-y z$ at $x=2 / 3, y=2 / 9,$ and $z=-3 / 5$

67) $-x^{2}$ at $x=-8 / 3$

Answer

$-\dfrac{64}{9}$

68) $-x^{4}$ at $x=-9 / 7$

69) $x^{2}+y z$ at $x=7 / 2, y=-5 / 4,$ and $z=-5 / 3$

Answer

$\dfrac{43}{3}$

70) $x^{2}+y z$ at $x=1 / 2, y=7 / 8,$ and $z=-5 / 9$

71) $a + b/c + d$ is equivalent to which of the following mathematical expressions?

1. $a+\dfrac{b}{c}+d$
2. $\dfrac{a+b}{c+d}$
3. $\dfrac{a+b}{c}+d$
4. $a+\dfrac{b}{c+d}$

Answer

(a)

72) $( a+b)/c+d$ is equivalent to which of the following mathematical expressions?

1. $a+\dfrac{b}{c}+d$
2. $\dfrac{a+b}{c+d}$
3. $\dfrac{a+b}{c}+d$
4. $a+\dfrac{b}{c+d}$

73) $a +b/(c+d)$ is equivalent to which of the following mathematical expressions?

1. $a+\dfrac{b}{c}+d$
2. $\dfrac{a+b}{c+d}$
3. $\dfrac{a+b}{c}+d$
4. $a+\dfrac{b}{c+d}$

Answer

(d)

74) ( a + b)/(c + d) is equivalent to which of the following mathematical expressions?

1. $a+\dfrac{b}{c}+d$
2. $\dfrac{a+b}{c+d}$
3. $\dfrac{a+b}{c}+d$
4. $a+\dfrac{b}{c+d}$

75) Use the graphing calculator to reduce $4125/1155$ to lowest terms.

Answer

$\dfrac{25}{7}$

76) Use the graphing calculator to reduce $2100/945$ to lowest terms.

77) Use the graphing calculator to simplify: $\dfrac{45}{84} \cdot \dfrac{70}{33}$

Answer

$\dfrac{25}{22}$

78) Use the graphing calculator to simplify: $\dfrac{34}{55}+\dfrac{13}{77}$

79) Use the graphing calculator to simplify: $-\dfrac{28}{33} \div\left(-\dfrac{35}{44}\right)$

Answer

$\dfrac{16}{15}$

80) Use the graphing calculator to simplify: $-\dfrac{11}{84}-\left(-\dfrac{11}{36}\right)$

1.4 Decimal Notation

In Exercises 1-33, simplify the given expressions.

1) $-2.835+(-8.759)$

Answer

$-11.594$

2) $-5.2+(-2)$

3) $19.5-(-1.6)$

Answer

21.1

4) $9.174-(-7.7)$

5) $-2-0.49$

Answer

$-2.49$

6) $-50.86-9$

7) $(-1.2)(-0.05)$

Answer

$0.06$

8) $(-7.9)(0.9)$

9) $-0.13+23.49$

Answer

$23.36$

10) $-30.82+75.93$

11) $16.4+(-41.205)$

Answer

$-24.805$

12) $-7.8+3.5$

13) $-0.4508 \div 0.49$

Answer

$-0.92$

14) $0.2378 \div(-0.29)$

15) $(-1.42)(-3.6)$

Answer

$5.112$

16) $(-8.64)(4.6)$

17) $2.184 \div(-0.24)$

Answer

$-9.1$

18) $7.395 \div(-0.87)$

19) $(-7.1)(-4.9)$

Answer

$34.79$

20) $(5.8)(-1.9)$

21) $7.41 \div(-9.5)$

Answer

$-0.78$

22) $-1.911 \div 4.9$

23) $-24.08 \div 2.8$

Answer

$-8.6$

24) $61.42 \div(-8.3)$

25) $(-4.04)(-0.6)$

Answer

$2.424$

26) $(-5.43)(0.09)$

27) $-7.2-(-7)$

Answer

$-0.2$

28) $-2.761-(-1.5)$

29) $(46.9)(-0.1)$

Answer

$-4.69$

30) $(-98.9)(-0.01)$

31) $(86.6)(-1.9)$

Answer

$-164.54$

32) $(-20.5)(8.1)$

In Exercises 33-60, simplify the given expression.

33) $-4.3-(-6.1)(-2.74)$

Answer

$-21.014$

34) $-1.4-1.9(3.36)$

35) $-3.49+|-6.9-(-15.7)|$

Answer

$5.31$

36) $1.3+|-13.22-8.79|$

37) $|18.9-1.55|-|-16.1-(-17.04)|$

Answer

$16.41$

38) $|-17.5-16.4|-|-15.58-(-4.5)|$

39) $8.2-(-3.1)^{3}$

Answer

$37.991$

40) $-8.4-(-6.8)^{3}$

41) $5.7-(-8.6)(1.1)^{2}$

Answer

$16.106$

42) $4.8-6.3(6.4)^{2}$

43) $(5.67)(6.8)-(1.8)^{2}$

Answer

$35.316$

44) $(-8.7)(8.3)-(-1.7)^{2}$

45) $9.6+(-10.05-13.16)$

Answer

$-13.61$

46) $-4.2+(17.1-14.46)$

47) $8.1+3.7(5.77)$

Answer

$29.449$

48) $8.1+2.3(-5.53)$

49) $7.5+34.5 /(-1.6+8.5)$

Answer

$12.5$

50) $-8.8+0.3 /(-7.2+7.3)$

51) $(8.0+2.2) / 5.1-4.6$

Answer

$\(-2.6$\)

52) $(35.3+1.8) / 5.3-5.4$

53) $-18.24-|-18.5-19.7|$

Answer

$-56.44$

54) $16.8-|4.58-17.14|$

55) $-4.37-|-8.97|$

Answer

$-13.34$

56) $4.1-|-8.4|$

57) $7.06-(-1.1-4.41)$

Answer

$12.57$

58) $7.74-(0.9-7.37)$

59) $-2.2-(-4.5)^{2}$

Answer

$-22.45$

60) $-2.8-(-4.3)^{2}$

61) Evaluate $a−b^2$ at $a =−2.9$ and $b =−5.4$.

Answer

$-32.06$

62) Evaluate $a−b^3$ at $a =−8.3$ and $b =−6.9$.

63) Evaluate $a+|b−c|$ at $a =−19.55$, $b =5.62$, and $c = −5.21$.

Answer

$-8.72$

64) Evaluate $a −| b − c|$ at $a = −8.37$, $b = −8.31$, and $c = 17.5$.

65) Evaluate $a−bc$ at $a =4 .3$, $b =8 .5$, and $c =1 .73$.

Answer

$-10.405$

66) Evaluate $a + bc$ at $a =4 .1$, $b =3.1$, and $c =−7.03$.

67) Evaluate $a − (b − c)$ at $a = −7.36$, $b = −17.6$, and $c = −19.07$.

Answer

$-8.83$

68) Evaluate $|a- b|−| c − d|$ at $a =1 .91$, $b = 19.41$, $c = −11.13$, and $d = 4.3$.

69) Evaluate $a+b/(c+d)$ at $a =4.7$, $b = 54.4$, $c =1.7$, and $d =5.1$.

Answer

$12.7$

70) Evaluate $(a + b)/c − d$ at $a = −74.2$, $b =3.8$, $c =8.8$, and $d =7.5$.

71) Evaluate $ab −c^2$ at $a = −2.45$, $b =5.6$, and $c =−3.2$.

Answer

$-23.96$

72) Evaluate $a +( b − c)$ at $a = 12 .6$, $b = −13.42$, and $c = −15.09$.

73) Evaluate $a−|b|$ at $a =−4.9$ and $b =−2.67$.

Answer

$-7.57$

74) Evaluate $a−bc^2$ at $a = −3.32$, $b = −5.4$, and $c =−8.5$.

75) Use your graphing calculator to evaluate $3.5−1.7x$ at $x =1 .25$. Round your answer to the nearest tenth.

Answer

$1.4$

76) Use your graphing calculator to evaluate $2.35x−1.7$ at $x = −12.23$. Round your answer to the nearest tenth.

77) Use your graphing calculator to evaluate $1.7x^2−3.2x+4.5$ at $x =2.86$. Round your answer to the nearest hundredth.

Answer

$9.25$

78) Use your graphing calculator to evaluate $19.5−4.4x−1.2x^2$ at $x = −1.23$. Round your answer to the nearest hundredth.

79) Use your graphing calculator to evaluate $−18.6+4.4x^2 −3.2x^3$ at $x =1.27$. Round your answer to the nearest thousandth.

Answer

$-4.948$

80) Use your graphing calculator to evaluate $−4.4x^3−7.2x−18.2$ at $x =2.29$. Round your answer to the nearest thousandth.

1.5 Algebraic Expressions

In Exercises 1-6, use the associative property of multiplication to simplify the expression.

Note: You must show the regrouping step using the associative property on your homework.

1) $-3(6 a)$

Answer

$-18 a$

2) $-10(2 y)$

3) $-9(6 a b)$

Answer

$-54 a b$

4) 8$(5 x y)$

5) $-7\left(3 x^{2}\right)$

Answer

$-21 x^{2}$

6) $-6(8 z)$

In Exercises 7-18, use the distributive property to expand the given expression.

7) 4$(3 x-7 y)$

Answer

$12 x-28 y$

8) $-4(5 a+2 b)$

9) $-6(-y+9)$

Answer

$6 y-54$

10) 5$(-9 w+6)$

11) $-9(s+9)$

Answer

$-9 s-81$

12) 6$(-10 y+3)$

13) $-(-3 u-6 v+8)$

Answer

$3 u+6 v-8$

14) $-(3 u-3 v-9)$

15) $-8\left(4 u^{2}-6 v^{2}\right)$

Answer

$-32 u^{2}+48 v^{2}$

16) $-5(8 x-9 y)$

17) $-(7 u+10 v+8)$

Answer

$-7 u-10 v-8$

18) $-(7 u-8 v-5)$

In Exercises 19-26, combine like terms by first using the distributive property to factor out the common variable part, and then simplifying.

Note: You must show the factoring step on your homework.

19) $-19 x+17 x-17 x$

Answer

$-19 x$

20) $11 n-3 n-18 n$

21) $14 x^{3}-10 x^{3}$

Answer

$4 x^{3}$

22) $-11 y^{3}-6 y^{3}$

23) $9 y^{2} x+13 y^{2} x-3 y^{2} x$

Answer

$19 y^{2} x$

24) $4 x^{3}-8 x^{3}+16 x^{3}$

25) $15 m+14 m$

Answer

$29 m$

26) $19 q+5 q$

In Exercises 27-38, simplify each of the following expressions by rearranging and combining like terms mentally.

Note: This means write down the problem, then write down the answer. No work.

27) $9-17 m-m+7$

Answer

$16-18 m$

28) $-11+20 x+16 x-14$

29) $-6 y^{2}-3 x^{3}+4 y^{2}+3 x^{3}$

Answer

$-2 y^{2}$

30) $14 y^{3}-11 y^{2} x+11 y^{3}+10 y^{2} x$

31) $-5 m-16+5-20 m$

Answer

$-25 m-11$

32) $-18 q+12-8-19 q$

33) $-16 x^{2} y+7 y^{3}-12 y^{3}-12 x^{2} y$

Answer

$-28 x^{2} y-5 y^{3}$

34) $10 x^{3}+4 y^{3}-13 y^{3}-14 x^{3}$

35) $-14 r+16-7 r-17$

Answer

$-21 r-1$

36) $-9 s-5-10 s+15$

37) $14-16 y-10-13 y$

Answer

$4-29 y$

38) $18+10 x+3-18 x$

In Exercises 39-58, use the distributive property to expand the expression, then combine like terms mentally.

39) $3-(-5 y+1)$

Answer

$2+5 y$

40) $5-(-10 q+3)$

41) $-\left(9 y^{2}+2 x^{2}\right)-8\left(5 y^{2}-6 x^{2}\right)$

Answer

$-49 y^{2}+46 x^{2}$

42) $-8\left(-8 y^{2}+4 x^{3}\right)-7\left(3 y^{2}+x^{3}\right)$

43) $2(10-6 p)+10(-2 p+5)$

Answer

$70-32 p$

44) $2(3-7 x)+(-7 x+9)$

45) $4(-10 n+5)-7(7 n-9)$

Answer

$-89 n+83$

46) $3(-9 n+10)+6(-7 n+8)$

47) $-4 x-4-(10 x-5)$

Answer

$-14 x+1$

48) $8 y+9-(-8 y+8)$

49) $-7-(5+3 x)$

Answer

$-12-3 x$

50) $10-(6-4 m)$

51) $-8(-5 y-8)-7(-2+9 y)$

Answer

$-23 y+78$

52) $6(-3 s+7)-(4-2 s)$

53) $4\left(-7 y^{2}-9 x^{2} y\right)-6\left(-5 x^{2} y-5 y^{2}\right)$

Answer

$2 y^{2}-6 x^{2} y$

54) $-6\left(x^{3}+3 y^{2} x\right)+8\left(-y^{2} x-9 x^{3}\right)$

55) $6 s-7-(2-4 s)$

Answer

$10 s-9$

56) $4 x-9-(-6+5 x)$

57) $9(9-10 r)+(-8-2 r)$

Answer

$73-92 r$

58) $-7(6+2 p)+5(5-5 p)$

In Exercises 59-64, use the distributive property to simplify the given expression.

59) $-7 x+7(2 x-5[8 x+5])$

Answer

$-273 x-175$

60) $-9 x+2(5 x+6[-8 x-3])$

61) $6 x-4(-3 x+2[5 x-7])$

Answer

$-22 x+56$

62) $2 x+4(5 x-7[8 x+9])$

63) $-8 x-5(2 x-3[-4 x+9])$

Answer

$-78 x+135$

64) $8 x+6(3 x+7[-9 x+5])$

Contributors and Attributions

• David Arnold (Retired Professor (Mathematics) at College of the Redwoods)