
# 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) $$|5|$$

$$5$$

2) $$|1|$$

3) $$|-2|$$

$$2$$

4) $$|-1|$$

5) $$|2|$$

$$2$$

6) $$|8|$$

7) $$|-4|$$

$$4$$

8) $$|-6|$$

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

9) $$-91+(-147)$$

$$-238$$

10) $$-23+(-13)$$

11) $$96+145$$

$$241$$

12) $$16+127$$

13) $$-76+46$$

$$-30$$

14) $$-11+21$$

15) $$-59+(-12)$$

$$-71$$

16) $$-40+(-58)$$

17) $$37+(-86)$$

$$-49$$

18) $$143+(-88)$$

19) $$66+(-85)$$

$$-19$$

20) $$33+(-41)$$

21) $$57+20$$

$$77$$

22) $$66+110$$

23) $$-48+127$$

$$79$$

24) $$-48+92$$

In Exercises 25-32, ﬁnd the diﬀerence.

25) $$-20-(-10)$$

$$-10$$

26) $$-20-(-20)$$

27) $$-62-7$$

$$-69$$

28) $$-82-62$$

29) $$-77-26$$

$$-103$$

30) $$-96-92$$

31) $$-7-(-16)$$

$$9$$

32) $$-20-(-5)$$

In Exercises 33-40, compute the exact value.

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

$$262144$$

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

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

$$−16807$$

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

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

$$81$$

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

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

$$256$$

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

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

41) $$-562-1728$$

$$−2290$$

42) $$-3125-(-576)$$

43) $$-400-(-8225)$$

$$7825$$

44) $$-8176+578$$

45) $$(-856)(232)$$

$$−198592$$

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

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

$$2916885$$

48) $$(753)(-9753)$$

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

$$−5832$$

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

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

$$−371293$$

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

## 1.2 Order of Operations

In Exercises 1-18, simplify the given expression.

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

$$-36$$

2) $$11+11(7)$$

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

$$32$$

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

5) $$-|-40|$$

$$-40$$

6) $$-|-42|$$

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

$$-4$$

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

9) $$-(-50)$$

$$50$$

10) $$-(-30)$$

11) $$-3^{5}$$

$$-243$$

12) $$-3^{2}$$

13) $$48 \div 4(6)$$

$$72$$

14) $$96 \div 6(4)$$

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

$$12$$

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

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

$$16$$

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

In Exercises 19-42, simplify the given expression.

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

$$-3$$

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

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

$$7$$

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

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

$$−324$$

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

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

$$-27$$

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

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

$$-4$$

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

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

$$24$$

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

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

$$4$$

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

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

$$31$$

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

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

$$43$$

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

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

$$-13$$

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

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

$$-19$$

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

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

$$40$$

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 $$y=0$$

$$36$$

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

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

$$81$$

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

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

$$-125$$

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

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

$$86$$

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

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

$$6$$

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

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

$$−21$$

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

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

$$−7$$

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

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

$$19$$

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 $$b =−30$$.

$$-543$$

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 $$d = 43$$.

$$1$$

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 $$d = 11$$.

$$-1$$

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?

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?

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

$$167920$$

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

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

$$-5$$

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 $$(A^2+B^2)/(A+B)$$.

$$−1745$$

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.

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

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.

$$8750$$ 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) $$\dfrac{20}{50}$$

$$\dfrac{2}{5}$$

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

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

$$\dfrac{5}{24}$$

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

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

$$\dfrac{8}{15}$$

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

$$\dfrac{9}{10}$$

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

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

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

$$\dfrac{4}{3}$$

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

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

$$\dfrac{3}{2}$$

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

$$-\dfrac{45}{13}$$

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

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

$$\dfrac{171}{26}$$

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

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

$$-\dfrac{19}{3}$$

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

$$\dfrac{10}{49}$$

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

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

$$-\dfrac{126}{275}$$

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

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

$$-\dfrac{270}{29}$$

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

$$-\dfrac{580}{39}$$

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

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

$$-\dfrac{930}{289}$$

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

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

$$\dfrac{98}{65}$$

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

$$-\dfrac{7}{12}$$

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

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

$$-\dfrac{11}{9}$$

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

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

$$-\dfrac{1}{36}$$

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

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

$$-\dfrac{76}{45}$$

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

In Exercises 39-52, simplify the expression.

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

$$-\dfrac{109}{90}$$

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

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

$$\dfrac{79}{36}$$

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

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

$$\dfrac{53}{35}$$

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

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

$$-\dfrac{131}{8}$$

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

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

$$-\dfrac{323}{50}$$

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

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

$$\dfrac{62}{45}$$

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

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

$$-\dfrac{5}{42}$$

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 $$z=5 / 2$$

$$-\dfrac{73}{12}$$

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 $$y=-1 / 2$$

$$-\dfrac{37}{16}$$

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 $$y=-3 / 4$$

$$\dfrac{81}{16}$$

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 $$z=2 / 5$$

$$-\dfrac{2}{5}$$

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 $$c=-5 / 2$$

$$-\dfrac{43}{10}$$

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

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

$$-\dfrac{1}{8}$$

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

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

$$-\dfrac{16}{15}$$

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

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

$$-\dfrac{64}{9}$$

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

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

$$\dfrac{43}{3}$$

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. $$a+\dfrac{b}{c+d}$$

(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. $$a+\dfrac{b}{c+d}$$

(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.

$$\dfrac{25}{7}$$

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

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

$$\dfrac{25}{22}$$

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

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

$$\dfrac{16}{15}$$

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) $$-2.835+(-8.759)$$

$$-11.594$$

2) $$-5.2+(-2)$$

3) $$19.5-(-1.6)$$

21.1

4) $$9.174-(-7.7)$$

5) $$-2-0.49$$

$$-2.49$$

6) $$-50.86-9$$

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

$$0.06$$

8) $$(-7.9)(0.9)$$

9) $$-0.13+23.49$$

$$23.36$$

10) $$-30.82+75.93$$

11) $$16.4+(-41.205)$$

$$-24.805$$

12) $$-7.8+3.5$$

13) $$-0.4508 \div 0.49$$

$$-0.92$$

14) $$0.2378 \div(-0.29)$$

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

$$5.112$$

16) $$(-8.64)(4.6)$$

17) $$2.184 \div(-0.24)$$

$$-9.1$$

18) $$7.395 \div(-0.87)$$

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

$$34.79$$

20) $$(5.8)(-1.9)$$

21) $$7.41 \div(-9.5)$$

$$-0.78$$

22) $$-1.911 \div 4.9$$

23) $$-24.08 \div 2.8$$

$$-8.6$$

24) $$61.42 \div(-8.3)$$

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

$$2.424$$

26) $$(-5.43)(0.09)$$

27) $$-7.2-(-7)$$

$$-0.2$$

28) $$-2.761-(-1.5)$$

29) $$(46.9)(-0.1)$$

$$-4.69$$

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

31) $$(86.6)(-1.9)$$

$$-164.54$$

32) $$(-20.5)(8.1)$$

In Exercises 33-60, simplify the given expression.

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

$$-21.014$$

34) $$-1.4-1.9(3.36)$$

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

$$5.31$$

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

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

$$16.41$$

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

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

$$37.991$$

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

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

$$16.106$$

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

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

$$35.316$$

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

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

$$-13.61$$

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

47) $$8.1+3.7(5.77)$$

$$29.449$$

48) $$8.1+2.3(-5.53)$$

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

$$12.5$$

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

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

$$\(-2.6$$\)

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

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

$$-56.44$$

54) $$16.8-|4.58-17.14|$$

55) $$-4.37-|-8.97|$$

$$-13.34$$

56) $$4.1-|-8.4|$$

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

$$12.57$$

58) $$7.74-(0.9-7.37)$$

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

$$-22.45$$

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

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

$$-32.06$$

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 $$c = −5.21$$.

$$-8.72$$

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 $$c =1 .73$$.

$$-10.405$$

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 $$c = −19.07$$.

$$-8.83$$

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 $$d =5.1$$.

$$12.7$$

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 $$c =−3.2$$.

$$-23.96$$

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 $$b =−2.67$$.

$$-7.57$$

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.

$$1.4$$

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.

$$9.25$$

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.

$$-4.948$$

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) $$-3(6 a)$$

$$-18 a$$

2) $$-10(2 y)$$

3) $$-9(6 a b)$$

$$-54 a b$$

4) 8$$(5 x y)$$

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

$$-21 x^{2}$$

6) $$-6(8 z)$$

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

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

$$12 x-28 y$$

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

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

$$6 y-54$$

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

11) $$-9(s+9)$$

$$-9 s-81$$

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

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

$$3 u+6 v-8$$

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

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

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

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

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

$$-7 u-10 v-8$$

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

$$-19 x$$

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

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

$$4 x^{3}$$

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

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

$$19 y^{2} x$$

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

25) $$15 m+14 m$$

$$29 m$$

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

$$16-18 m$$

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

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

$$-2 y^{2}$$

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

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

$$-25 m-11$$

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

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

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

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

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

$$-21 r-1$$

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

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

$$4-29 y$$

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

$$2+5 y$$

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

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

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

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

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

$$70-32 p$$

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

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

$$-89 n+83$$

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

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

$$-14 x+1$$

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

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

$$-12-3 x$$

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

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

$$-23 y+78$$

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

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

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

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

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

$$10 s-9$$

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

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

$$73-92 r$$

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

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

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

$$-273 x-175$$

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

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

$$-22 x+56$$

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

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

$$-78 x+135$$
64) $$8 x+6(3 x+7[-9 x+5])$$