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)