# Exercises: Divide Monomials (5.5)

## Exercises: Simplify Expressions Using the Quotient Property for Exponents

Instructions: For questions 1-8, simplify.

1.

a. $\frac{{x}^{18}}{{x}^{3}}$
b. $\frac{{5}^{12}}{{5}^{3}}$

2.

a. $\frac{{y}^{20}}{{y}^{10}}$
b. $\frac{{7}^{16}}{{7}^{2}}$

Solution

a. ${y}^{10}$
b. ${7}^{14}$

3.

a. $\frac{{p}^{21}}{{p}^{7}}$
b. $\frac{{4}^{16}}{{4}^{4}}$

4.

a. $\frac{{u}^{24}}{{u}^{3}}$
b. $\frac{{9}^{15}}{{9}^{5}}$

Solution

a. ${u}^{21}$
b. ${9}^{10}$

5.

a. $\frac{{q}^{18}}{{q}^{36}}$
b. $\frac{{10}^{2}}{{10}^{3}}$

6.

a. $\frac{{t}^{10}}{{t}^{40}}$
b. $\frac{{8}^{3}}{{8}^{5}}$

Solution

a. $\frac{1}{{t}^{30}}$
b. $\frac{1}{64}$

7.

a. $\frac{b}{{b}^{9}}$
b. $\frac{4}{{4}^{6}}$

8.

a. $\frac{x}{{x}^{7}}$
b. $\frac{10}{{10}^{3}}$

Solution

a. $\frac{1}{{x}^{6}}$
b. $\frac{1}{100}$

## Exercises: Simplify Expressions with Zero Exponents

Instructions: For questions 9-18, simplify.

9.

a. ${20}^{0}$
b. ${b}^{0}$

10.

a. ${13}^{0}$
b. ${k}^{0}$

Solution

a. $1$
b. $1$

11.

a. $-{27}^{0}$
b. $-\left({27}^{0}\right)$

12.

a. $-{15}^{0}$
b. $-\left({15}^{0}\right)$

Solution

a. $-1$
b. $-1$

13.

a. ${\left(25x\right)}^{0}$
b. $25{x}^{0}$

14.

a. ${\left(6y\right)}^{0}$
b. $6{y}^{0}$

Solution

a. $1$
b. $6$

15.

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

16.

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

Solution

a. $7$
b. $1$

17.

a. $12{n}^{0}-18{m}^{0}$
b. ${\left(12n\right)}^{0}-{\left(18m\right)}^{0}$

18.

a. $15{r}^{0}-22{s}^{0}$
b. ${\left(15r\right)}^{0}-{\left(22s\right)}^{0}$

Solution

a. $-7$
b. $0$

## Exercises: Simplify Expressions Using the Quotient to a Power Property

Instructions: For questions 19-22, simplify.

19.

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

20.

a. ${\left(\frac{2}{5}\right)}^{2}$
b. ${\left(\frac{x}{3}\right)}^{4}$
c. ${\left(\frac{a}{b}\right)}^{5}$

Solution

a. $\frac{4}{25}$
b. $\frac{{x}^{4}}{81}$
c. $\frac{{a}^{5}}{{b}^{5}}$

21.

a. ${\left(\frac{a}{3b}\right)}^{4}$
b. ${\left(\frac{5}{4m}\right)}^{2}$

22.

a. ${\left(\frac{x}{2y}\right)}^{3}$
b. ${\left(\frac{10}{3q}\right)}^{4}$

Solution

a. $\frac{{x}^{3}}{8{y}^{3}}$
b. $\frac{10,000}{81{q}^{4}}$

## Exercises: Simplify Expressions by Applying Several Properties

Instructions: For questions 23-50, simplify.

23. $\frac{{\left({a}^{2}\right)}^{3}}{{a}^{4}}$

24. $\frac{{\left({p}^{3}\right)}^{4}}{{p}^{5}}$
Solution

${p}^{7}$

25. $\frac{{\left({y}^{3}\right)}^{4}}{{y}^{10}}$

26. $\frac{{\left({x}^{4}\right)}^{5}}{{x}^{15}}$
Solution

${x}^{5}$

27. $\frac{{u}^{6}}{{\left({u}^{3}\right)}^{2}}$

28. $\frac{{v}^{20}}{{\left({v}^{4}\right)}^{5}}$
Solution

$1$

29. $\frac{{m}^{12}}{{\left({m}^{8}\right)}^{3}}$

30. $\frac{{n}^{8}}{{\left({n}^{6}\right)}^{4}}$
Solution

$\frac{1}{{n}^{16}}$

31. ${\left(\frac{{p}^{9}}{{p}^{3}}\right)}^{5}$

32. ${\left(\frac{{q}^{8}}{{q}^{2}}\right)}^{3}$
Solution

${q}^{18}$

33. ${\left(\frac{{r}^{2}}{{r}^{6}}\right)}^{3}$

34. ${\left(\frac{{m}^{4}}{{m}^{7}}\right)}^{4}$
Solution

$\frac{1}{{m}^{12}}$

35. ${\left(\frac{p}{{r}^{11}}\right)}^{2}$

36. ${\left(\frac{a}{{b}^{6}}\right)}^{3}$
Solution

$\frac{{a}^{3}}{{b}^{18}}$

37. ${\left(\frac{{w}^{5}}{{x}^{3}}\right)}^{8}$

38. ${\left(\frac{{y}^{4}}{{z}^{10}}\right)}^{5}$
Solution

$\frac{{y}^{20}}{{z}^{50}}$

39. ${\left(\frac{2{j}^{3}}{3k}\right)}^{4}$

40. ${\left(\frac{3{m}^{5}}{5n}\right)}^{3}$
Solution

$\frac{27{m}^{15}}{125{n}^{3}}$

41. ${\left(\frac{3{c}^{2}}{4{d}^{6}}\right)}^{3}$

42. ${\left(\frac{5{u}^{7}}{2{v}^{3}}\right)}^{4}$
Solution

$\frac{625{u}^{28}}{16{v}^{{}^{12}}}$

43. ${\left(\frac{{k}^{2}{k}^{8}}{{k}^{3}}\right)}^{2}$

44. ${\left(\frac{{j}^{2}{j}^{5}}{{j}^{4}}\right)}^{3}$
Solution

${j}^{9}$

45. $\frac{{\left({t}^{2}\right)}^{5}{\left({t}^{4}\right)}^{2}}{{\left({t}^{3}\right)}^{7}}$

46. $\frac{{\left({q}^{3}\right)}^{6}{\left({q}^{2}\right)}^{3}}{{\left({q}^{4}\right)}^{8}}$
Solution

$\frac{1}{{q}^{8}}$

47. $\frac{{\left(-2{p}^{2}\right)}^{4}{\left(3{p}^{4}\right)}^{2}}{{\left(-6{p}^{3}\right)}^{2}}$

48. $\frac{{\left(-2{k}^{3}\right)}^{2}{\left(6{k}^{2}\right)}^{4}}{{\left(9{k}^{4}\right)}^{2}}$
Solution

$64{k}^{6}$

49. $\frac{{\left(-4{m}^{3}\right)}^{2}{\left(5{m}^{4}\right)}^{3}}{{\left(-10{m}^{6}\right)}^{3}}$

50. $\frac{{\left(-10{n}^{2}\right)}^{3}{\left(4{n}^{5}\right)}^{2}}{{\left(2{n}^{8}\right)}^{2}}$
Solution

$-4\text{,}000$

## Exercises: Divide Monomials

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

51. $56{b}^{8}\div 7{b}^{2}$

52. $63{v}^{10}\div 9{v}^{2}$
Solution

$7{v}^{8}$

53. $-88{y}^{15}\div 8{y}^{3}$

54. $-72{u}^{12}\div 12{u}^{4}$
Solution

$-6{u}^{8}$

55. $\frac{45{a}^{6}{b}^{8}}{-15{a}^{10}{b}^{2}}$

56. $\frac{54{x}^{9}{y}^{3}}{-18{x}^{6}{y}^{15}}$
Solution

$-\frac{3{x}^{3}}{{y}^{12}}$

57. $\frac{15{r}^{4}{s}^{9}}{18{r}^{9}{s}^{2}}$

58. $\frac{20{m}^{8}{n}^{4}}{30{m}^{5}{n}^{9}}$
Solution

$\frac{-2{m}^{3}}{3{n}^{5}}$

59. $\frac{18{a}^{4}{b}^{8}}{-27{a}^{9}{b}^{5}}$

60. $\frac{45{x}^{5}{y}^{9}}{-60{x}^{8}{y}^{6}}$
Solution

$\frac{-3{y}^{3}}{4{x}^{3}}$

61. $\frac{64{q}^{11}{r}^{9}{s}^{3}}{48{q}^{6}{r}^{8}{s}^{5}}$

62. $\frac{65{a}^{10}{b}^{8}{c}^{5}}{42{a}^{7}{b}^{6}{c}^{8}}$
Solution

$\frac{65{a}^{3}{b}^{2}}{42{c}^{3}}$

63. $\frac{\left(10{m}^{5}{n}^{4}\right)\left(5{m}^{3}{n}^{6}\right)}{25{m}^{7}{n}^{5}}$

64. $\frac{\left(-18{p}^{4}{q}^{7}\right)\left(-6{p}^{3}{q}^{8}\right)}{-36{p}^{12}{q}^{10}}$
Solution

$\frac{-3{q}^{5}}{{p}^{5}}$

65. $\frac{\left(6{a}^{4}{b}^{3}\right)\left(4a{b}^{5}\right)}{\left(12{a}^{2}b\right)\left({a}^{3}b\right)}$

66. $\frac{\left(4{u}^{2}{v}^{5}\right)\left(15{u}^{3}v\right)}{\left(12{u}^{3}v\right)\left({u}^{4}v\right)}$
Solution

$\frac{5{v}^{4}}{{u}^{2}}$

## Exercises: Mixed Practice

Instructions: For questions 67-80, solve.

67.

a. $24{a}^{5}+2{a}^{5}$
b. $24{a}^{5}-2{a}^{5}$
c. $24{a}^{5}\cdot 2{a}^{5}$
d. $24{a}^{5}\div 2{a}^{5}$

68.

a. $15{n}^{10}+3{n}^{10}$
b. $15{n}^{10}-3{n}^{10}$
c. $15{n}^{10}\cdot 3{n}^{10}$
d. $15{n}^{10}\div 3{n}^{10}$

Solution

a. $18{n}^{10}$
b. $12{n}^{10}$
c. $45{n}^{20}$
d. $5$

69.

a. ${p}^{4}\cdot {p}^{6}$
b. ${\left({p}^{4}\right)}^{6}$

70.

a. ${q}^{5}\cdot {q}^{3}$
b. ${\left({q}^{5}\right)}^{3}$

Solution

a. ${q}^{8}$
b. ${q}^{15}$

71.

a. $\frac{{y}^{3}}{y}$
b. $\frac{y}{{y}^{3}}$

72.

a. $\frac{{z}^{6}}{{z}^{5}}$
b. $\frac{{z}^{5}}{{z}^{6}}$

Solution

a. $z$
b. $\frac{1}{z}$

73. $\left(8{x}^{5}\right)\left(9x\right)\div 6{x}^{3}$

74. $\left(4y\right)\left(12{y}^{7}\right)\div 8{y}^{2}$
Solution

$6{y}^{6}$

75. $\frac{27{a}^{7}}{3{a}^{3}}+\frac{54{a}^{9}}{9{a}^{5}}$

76. $\frac{32{c}^{11}}{4{c}^{5}}+\frac{42{c}^{9}}{6{c}^{3}}$
Solution

$15{c}^{6}$

77. $\frac{32{y}^{5}}{8{y}^{2}}-\frac{60{y}^{10}}{5{y}^{7}}$

78. $\frac{48{x}^{6}}{6{x}^{4}}-\frac{35{x}^{9}}{7{x}^{7}}$
Solution

$3{x}^{2}$

79. $\frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}-\frac{72{r}^{2}{s}^{2}}{6s}$

80. $\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}-\frac{45{y}^{2}{z}^{2}}{5y}$
Solution

$-y{z}^{2}$

## Exercises: Everyday Math

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

82. Memory. One gigabyte is approximately ${10}^{9}$ bytes. One terabyte is approximately ${10}^{12}$ bytes. How many gigabytes are in one terabyte?
Solution

${10}^{3}$

## Exercises: Writing Exercises

Instructions: For questions 83-86, answer the given writing exercises.
83. Jennifer thinks the quotient $\frac{{a}^{24}}{{a}^{6}}$ simplifies to ${a}^{4}$. What is wrong with her reasoning?

84. Maurice simplifies the quotient $\frac{{d}^{7}}{d}$ by writing $\frac{{\cancel{d}}^{7}}{\cancel{d}}=7$. What is wrong with his reasoning?
Solution

85. When Drake simplified $-{3}^{0}$ and ${\left(-3\right)}^{0}$ he got the same answer. Explain how using the Order of Operations correctly gives different answers.

86. Robert thinks ${x}^{0}$ simplifies to 0. What would you say to convince Robert he is wrong?
Solution