Exercises: Divide Monomials (5.5)

Exercises: Simplify Expressions Using the Quotient Property for Exponents

Instructions: For questions 1-8, simplify.

1.

a. [latex]\frac{{x}^{18}}{{x}^{3}}[/latex]
b. [latex]\frac{{5}^{12}}{{5}^{3}}[/latex]

2.

a. [latex]\frac{{y}^{20}}{{y}^{10}}[/latex]
b. [latex]\frac{{7}^{16}}{{7}^{2}}[/latex]

Solution

a. [latex]{y}^{10}[/latex]
b. [latex]{7}^{14}[/latex]

3.

a. [latex]\frac{{p}^{21}}{{p}^{7}}[/latex]
b. [latex]\frac{{4}^{16}}{{4}^{4}}[/latex]

4.

a. [latex]\frac{{u}^{24}}{{u}^{3}}[/latex]
b. [latex]\frac{{9}^{15}}{{9}^{5}}[/latex]

Solution

a. [latex]{u}^{21}[/latex]
b. [latex]{9}^{10}[/latex]

5.

a. [latex]\frac{{q}^{18}}{{q}^{36}}[/latex]
b. [latex]\frac{{10}^{2}}{{10}^{3}}[/latex]

6.

a. [latex]\frac{{t}^{10}}{{t}^{40}}[/latex]
b. [latex]\frac{{8}^{3}}{{8}^{5}}[/latex]

Solution

a. [latex]\frac{1}{{t}^{30}}[/latex]
b. [latex]\frac{1}{64}[/latex]

7.

a. [latex]\frac{b}{{b}^{9}}[/latex]
b. [latex]\frac{4}{{4}^{6}}[/latex]

8.

a. [latex]\frac{x}{{x}^{7}}[/latex]
b. [latex]\frac{10}{{10}^{3}}[/latex]

Solution

a. [latex]\frac{1}{{x}^{6}}[/latex]
b. [latex]\frac{1}{100}[/latex]

Exercises: Simplify Expressions with Zero Exponents

Instructions: For questions 9-18, simplify.

9.

a. [latex]{20}^{0}[/latex]
b. [latex]{b}^{0}[/latex]

10.

a. [latex]{13}^{0}[/latex]
b. [latex]{k}^{0}[/latex]

Solution

a. [latex]1[/latex]
b. [latex]1[/latex]

11.

a. [latex]-{27}^{0}[/latex]
b. [latex]-\left({27}^{0}\right)[/latex]

12.

a. [latex]-{15}^{0}[/latex]
b. [latex]-\left({15}^{0}\right)[/latex]

Solution

a. [latex]-1[/latex]
b. [latex]-1[/latex]

13.

a. [latex]{\left(25x\right)}^{0}[/latex]
b. [latex]25{x}^{0}[/latex]

14.

a. [latex]{\left(6y\right)}^{0}[/latex]
b. [latex]6{y}^{0}[/latex]

Solution

a. [latex]1[/latex]
b. [latex]6[/latex]

15.

a. [latex]{\left(12x\right)}^{0}[/latex]
b. [latex]{\left(-56{p}^{4}{q}^{3}\right)}^{0}[/latex]

16.

a. [latex]7{y}^{0}[/latex][latex]{\left(17y\right)}^{0}[/latex]
b. [latex]{\left(-93{c}^{7}{d}^{15}\right)}^{0}[/latex]

Solution

a. [latex]7[/latex]
b. [latex]1[/latex]

17.

a. [latex]12{n}^{0}-18{m}^{0}[/latex]
b. [latex]{\left(12n\right)}^{0}-{\left(18m\right)}^{0}[/latex]

18.

a. [latex]15{r}^{0}-22{s}^{0}[/latex]
b. [latex]{\left(15r\right)}^{0}-{\left(22s\right)}^{0}[/latex]

Solution

a. [latex]-7[/latex]
b. [latex]0[/latex]

Exercises: Simplify Expressions Using the Quotient to a Power Property

Instructions: For questions 19-22, simplify.

19.

a. [latex]{\left(\frac{3}{4}\right)}^{3}[/latex]
b. [latex]{\left(\frac{p}{2}\right)}^{5}[/latex]
c. [latex]{\left(\frac{x}{y}\right)}^{6}[/latex]

20.

a. [latex]{\left(\frac{2}{5}\right)}^{2}[/latex]
b. [latex]{\left(\frac{x}{3}\right)}^{4}[/latex]
c. [latex]{\left(\frac{a}{b}\right)}^{5}[/latex]

Solution

a. [latex]\frac{4}{25}[/latex]
b. [latex]\frac{{x}^{4}}{81}[/latex]
c. [latex]\frac{{a}^{5}}{{b}^{5}}[/latex]

21.

a. [latex]{\left(\frac{a}{3b}\right)}^{4}[/latex]
b. [latex]{\left(\frac{5}{4m}\right)}^{2}[/latex]

22.

a. [latex]{\left(\frac{x}{2y}\right)}^{3}[/latex]
b. [latex]{\left(\frac{10}{3q}\right)}^{4}[/latex]

Solution

a. [latex]\frac{{x}^{3}}{8{y}^{3}}[/latex]
b. [latex]\frac{10,000}{81{q}^{4}}[/latex]

Exercises: Simplify Expressions by Applying Several Properties

Instructions: For questions 23-50, simplify.

23. [latex]\frac{{\left({a}^{2}\right)}^{3}}{{a}^{4}}[/latex]
24. [latex]\frac{{\left({p}^{3}\right)}^{4}}{{p}^{5}}[/latex]
Solution

[latex]{p}^{7}[/latex]

25. [latex]\frac{{\left({y}^{3}\right)}^{4}}{{y}^{10}}[/latex]
26. [latex]\frac{{\left({x}^{4}\right)}^{5}}{{x}^{15}}[/latex]
Solution

[latex]{x}^{5}[/latex]

27. [latex]\frac{{u}^{6}}{{\left({u}^{3}\right)}^{2}}[/latex]
28. [latex]\frac{{v}^{20}}{{\left({v}^{4}\right)}^{5}}[/latex]
Solution

[latex]1[/latex]

29. [latex]\frac{{m}^{12}}{{\left({m}^{8}\right)}^{3}}[/latex]
30. [latex]\frac{{n}^{8}}{{\left({n}^{6}\right)}^{4}}[/latex]
Solution

[latex]\frac{1}{{n}^{16}}[/latex]

31. [latex]{\left(\frac{{p}^{9}}{{p}^{3}}\right)}^{5}[/latex]
32. [latex]{\left(\frac{{q}^{8}}{{q}^{2}}\right)}^{3}[/latex]
Solution

[latex]{q}^{18}[/latex]

33. [latex]{\left(\frac{{r}^{2}}{{r}^{6}}\right)}^{3}[/latex]
34. [latex]{\left(\frac{{m}^{4}}{{m}^{7}}\right)}^{4}[/latex]
Solution

[latex]\frac{1}{{m}^{12}}[/latex]

35. [latex]{\left(\frac{p}{{r}^{11}}\right)}^{2}[/latex]
36. [latex]{\left(\frac{a}{{b}^{6}}\right)}^{3}[/latex]
Solution

[latex]\frac{{a}^{3}}{{b}^{18}}[/latex]

37. [latex]{\left(\frac{{w}^{5}}{{x}^{3}}\right)}^{8}[/latex]
38. [latex]{\left(\frac{{y}^{4}}{{z}^{10}}\right)}^{5}[/latex]
Solution

[latex]\frac{{y}^{20}}{{z}^{50}}[/latex]

39. [latex]{\left(\frac{2{j}^{3}}{3k}\right)}^{4}[/latex]
40. [latex]{\left(\frac{3{m}^{5}}{5n}\right)}^{3}[/latex]
Solution

[latex]\frac{27{m}^{15}}{125{n}^{3}}[/latex]

41. [latex]{\left(\frac{3{c}^{2}}{4{d}^{6}}\right)}^{3}[/latex]
42. [latex]{\left(\frac{5{u}^{7}}{2{v}^{3}}\right)}^{4}[/latex]
Solution

[latex]\frac{625{u}^{28}}{16{v}^{{}^{12}}}[/latex]

43. [latex]{\left(\frac{{k}^{2}{k}^{8}}{{k}^{3}}\right)}^{2}[/latex]
44. [latex]{\left(\frac{{j}^{2}{j}^{5}}{{j}^{4}}\right)}^{3}[/latex]
Solution

[latex]{j}^{9}[/latex]

45. [latex]\frac{{\left({t}^{2}\right)}^{5}{\left({t}^{4}\right)}^{2}}{{\left({t}^{3}\right)}^{7}}[/latex]
46. [latex]\frac{{\left({q}^{3}\right)}^{6}{\left({q}^{2}\right)}^{3}}{{\left({q}^{4}\right)}^{8}}[/latex]
Solution

[latex]\frac{1}{{q}^{8}}[/latex]

47. [latex]\frac{{\left(-2{p}^{2}\right)}^{4}{\left(3{p}^{4}\right)}^{2}}{{\left(-6{p}^{3}\right)}^{2}}[/latex]
48. [latex]\frac{{\left(-2{k}^{3}\right)}^{2}{\left(6{k}^{2}\right)}^{4}}{{\left(9{k}^{4}\right)}^{2}}[/latex]
Solution

[latex]64{k}^{6}[/latex]

49. [latex]\frac{{\left(-4{m}^{3}\right)}^{2}{\left(5{m}^{4}\right)}^{3}}{{\left(-10{m}^{6}\right)}^{3}}[/latex]
50. [latex]\frac{{\left(-10{n}^{2}\right)}^{3}{\left(4{n}^{5}\right)}^{2}}{{\left(2{n}^{8}\right)}^{2}}[/latex]
Solution

[latex]-4\text{,}000[/latex]

Exercises: Divide Monomials

Instructions: For questions 51-66, divide the monomials.

51. [latex]56{b}^{8}\div 7{b}^{2}[/latex]
52. [latex]63{v}^{10}\div 9{v}^{2}[/latex]
Solution

[latex]7{v}^{8}[/latex]

53. [latex]-88{y}^{15}\div 8{y}^{3}[/latex]
54. [latex]-72{u}^{12}\div 12{u}^{4}[/latex]
Solution

[latex]-6{u}^{8}[/latex]

55. [latex]\frac{45{a}^{6}{b}^{8}}{-15{a}^{10}{b}^{2}}[/latex]
56. [latex]\frac{54{x}^{9}{y}^{3}}{-18{x}^{6}{y}^{15}}[/latex]
Solution

[latex]-\frac{3{x}^{3}}{{y}^{12}}[/latex]

57. [latex]\frac{15{r}^{4}{s}^{9}}{18{r}^{9}{s}^{2}}[/latex]
58. [latex]\frac{20{m}^{8}{n}^{4}}{30{m}^{5}{n}^{9}}[/latex]
Solution

[latex]\frac{-2{m}^{3}}{3{n}^{5}}[/latex]

59. [latex]\frac{18{a}^{4}{b}^{8}}{-27{a}^{9}{b}^{5}}[/latex]
60. [latex]\frac{45{x}^{5}{y}^{9}}{-60{x}^{8}{y}^{6}}[/latex]
Solution

[latex]\frac{-3{y}^{3}}{4{x}^{3}}[/latex]

61. [latex]\frac{64{q}^{11}{r}^{9}{s}^{3}}{48{q}^{6}{r}^{8}{s}^{5}}[/latex]
62. [latex]\frac{65{a}^{10}{b}^{8}{c}^{5}}{42{a}^{7}{b}^{6}{c}^{8}}[/latex]
Solution

[latex]\frac{65{a}^{3}{b}^{2}}{42{c}^{3}}[/latex]

63. [latex]\frac{\left(10{m}^{5}{n}^{4}\right)\left(5{m}^{3}{n}^{6}\right)}{25{m}^{7}{n}^{5}}[/latex]
64. [latex]\frac{\left(-18{p}^{4}{q}^{7}\right)\left(-6{p}^{3}{q}^{8}\right)}{-36{p}^{12}{q}^{10}}[/latex]
Solution

[latex]\frac{-3{q}^{5}}{{p}^{5}}[/latex]

65. [latex]\frac{\left(6{a}^{4}{b}^{3}\right)\left(4a{b}^{5}\right)}{\left(12{a}^{2}b\right)\left({a}^{3}b\right)}[/latex]
66. [latex]\frac{\left(4{u}^{2}{v}^{5}\right)\left(15{u}^{3}v\right)}{\left(12{u}^{3}v\right)\left({u}^{4}v\right)}[/latex]
Solution

[latex]\frac{5{v}^{4}}{{u}^{2}}[/latex]

Exercises: Mixed Practice

Instructions: For questions 67-80, solve.

 67.

a. [latex]24{a}^{5}+2{a}^{5}[/latex]
b. [latex]24{a}^{5}-2{a}^{5}[/latex]
c. [latex]24{a}^{5}\cdot 2{a}^{5}[/latex]
d. [latex]24{a}^{5}\div 2{a}^{5}[/latex]

68.

a. [latex]15{n}^{10}+3{n}^{10}[/latex]
b. [latex]15{n}^{10}-3{n}^{10}[/latex]
c. [latex]15{n}^{10}\cdot 3{n}^{10}[/latex]
d. [latex]15{n}^{10}\div 3{n}^{10}[/latex]

Solution

a. [latex]18{n}^{10}[/latex]
b. [latex]12{n}^{10}[/latex]
c. [latex]45{n}^{20}[/latex]
d. [latex]5[/latex]

69.

a. [latex]{p}^{4}\cdot {p}^{6}[/latex]
b. [latex]{\left({p}^{4}\right)}^{6}[/latex]

70.

a. [latex]{q}^{5}\cdot {q}^{3}[/latex]
b. [latex]{\left({q}^{5}\right)}^{3}[/latex]

Solution

a. [latex]{q}^{8}[/latex]
b. [latex]{q}^{15}[/latex]

71.

a. [latex]\frac{{y}^{3}}{y}[/latex]
b. [latex]\frac{y}{{y}^{3}}[/latex]

72.

a. [latex]\frac{{z}^{6}}{{z}^{5}}[/latex]
b. [latex]\frac{{z}^{5}}{{z}^{6}}[/latex]

Solution

a. [latex]z[/latex]
b. [latex]\frac{1}{z}[/latex]

73. [latex]\left(8{x}^{5}\right)\left(9x\right)\div 6{x}^{3}[/latex]
74. [latex]\left(4y\right)\left(12{y}^{7}\right)\div 8{y}^{2}[/latex]
Solution

[latex]6{y}^{6}[/latex]

75. [latex]\frac{27{a}^{7}}{3{a}^{3}}+\frac{54{a}^{9}}{9{a}^{5}}[/latex]
76. [latex]\frac{32{c}^{11}}{4{c}^{5}}+\frac{42{c}^{9}}{6{c}^{3}}[/latex]
Solution

[latex]15{c}^{6}[/latex]

77. [latex]\frac{32{y}^{5}}{8{y}^{2}}-\frac{60{y}^{10}}{5{y}^{7}}[/latex]
78. [latex]\frac{48{x}^{6}}{6{x}^{4}}-\frac{35{x}^{9}}{7{x}^{7}}[/latex]
Solution

[latex]3{x}^{2}[/latex]

79. [latex]\frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}-\frac{72{r}^{2}{s}^{2}}{6s}[/latex]
80. [latex]\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}-\frac{45{y}^{2}{z}^{2}}{5y}[/latex]
Solution

[latex]-y{z}^{2}[/latex]

Exercises: Everyday Math

Instructions: For questions 81-82, answer the given everday math word problems.
81. Memory. One megabyte is approximately [latex]{10}^{6}[/latex] bytes. One gigabyte is approximately [latex]{10}^{9}[/latex] bytes. How many megabytes are in one gigabyte?
82. Memory. One gigabyte is approximately [latex]{10}^{9}[/latex] bytes. One terabyte is approximately [latex]{10}^{12}[/latex] bytes. How many gigabytes are in one terabyte?
Solution

[latex]{10}^{3}[/latex]

Exercises: Writing Exercises

Instructions: For questions 83-86, answer the given writing exercises.
83. Jennifer thinks the quotient [latex]\frac{{a}^{24}}{{a}^{6}}[/latex] simplifies to [latex]{a}^{4}[/latex]. What is wrong with her reasoning?
84. Maurice simplifies the quotient [latex]\frac{{d}^{7}}{d}[/latex] by writing [latex]\frac{{\cancel{d}}^{7}}{\cancel{d}}=7[/latex]. What is wrong with his reasoning?
Solution

Answers will vary.

85. When Drake simplified [latex]-{3}^{0}[/latex] and [latex]{\left(-3\right)}^{0}[/latex] he got the same answer. Explain how using the Order of Operations correctly gives different answers.
86. Robert thinks [latex]{x}^{0}[/latex] simplifies to 0. What would you say to convince Robert he is wrong?
Solution

Answers will vary.

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Fanshawe Pre-Health Sciences Mathematics 1 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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