6.5E: Exercises

Practice Makes Perfect

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

Exercise $\PageIndex{1}$

1. $\dfrac{x^{18}}{x^{3}}$
2. $\dfrac{5^{12}}{5^{3}}$

Exercise $\PageIndex{2}$

1. $\dfrac{y^{20}}{y^{10}}$
2. $\dfrac{7^{16}}{7^{2}}$

Answer

1. $y^{10}$
2. $7^{14}$

Exercise $\PageIndex{3}$

1. $\dfrac{p^{21}}{p^{7}}$
2. $\dfrac{4^{16}}{4^{4}}$

Exercise $\PageIndex{4}$

1. $\dfrac{u^{24}}{u^{3}}$
2. $\dfrac{9^{15}}{9^{5}}$

Answer

1. $u^{21}$
2. $9^{10}$

Exercise $\PageIndex{5}$

1. $\dfrac{q^{18}}{q^{36}}$
2. $\dfrac{10^{2}}{10^{3}}$

Exercise $\PageIndex{6}$

1. $\dfrac{t^{10}}{t^{40}}$
2. $\dfrac{8^{3}}{8^{5}}$

Answer

1. $\dfrac{1}{t^{30}}$
2. $\dfrac{1}{64}$

Exercise $\PageIndex{7}$

1. $\dfrac{b}{b^{9}}$
2. $\dfrac{4}{4^{6}}$

Exercise $\PageIndex{8}$

1. $\dfrac{x}{x^{7}}$
2. $\dfrac{10}{10^{3}}$

Answer

1. $\dfrac{1}{x^{6}}$
2. $\dfrac{1}{100}$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

Exercise $\PageIndex{9}$

1. $20^{0}$
2. $b^{0}$

Exercise $\PageIndex{10}$

1. $13^0$
2. $k^{0}$

Answer

1. 1
2. 1

Exercise $\PageIndex{11}$

1. $-27^{0}$
2. $-\left(27^{0}\right)$

Exercise $\PageIndex{12}$

1. $-15^{0}$
2. $-\left(15^{0}\right)$

Answer

1. −1
2. −1

Exercise $\PageIndex{13}$

1. $(25 x)^{0}$
2. $25 x^{0}$

Exercise $\PageIndex{14}$

1. $(6 y)^{0}$
2. $6 y^{0}$

Answer

1. 1
2. 6

Exercise $\PageIndex{15}$

1. $(12 x)^{0}$
2. $\left(-56 p^{4} q^{3}\right)^{0}$

Exercise $\PageIndex{16}$

1. 7$y^{0}(17 y)^{0}$
2. $\left(-93 c^{7} d^{15}\right)^{0}$

Answer

1. 7
2. 1

Exercise $\PageIndex{17}$

1. $12 n^{0}-18 m^{0}$
2. $(12 n)^{0}-(18 m)^{0}$

Exercise $\PageIndex{18}$

1. $15 r^{0}-22 s^{0}$
2. $(15 r)^{0}-(22 s)^{0}$

Answer

1. −7
2. 0

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Exercise $\PageIndex{19}$

1. $\left(\dfrac{3}{4}\right)^{3}$
2. $\left(\dfrac{p}{2}\right)^{5}$
3. $\left(\dfrac{x}{y}\right)^{6}$

Exercise $\PageIndex{20}$

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

Answer

1. $\dfrac{4}{25}$
2. $\dfrac{x^{4}}{81}$
3. $\left(\dfrac{a}{b}\right)^{5}$

Exercise $\PageIndex{21}$

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

Exercise $\PageIndex{22}$

1. $\left(\dfrac{a}{3 b}\right)^{4}$
2. $\left(\dfrac{10}{3 q}\right)^{4}$

Answer

1. $\dfrac{x^{3}}{8 y^{3}}$
2. $\dfrac{10,000}{81 q^{4}}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

Exercise $\PageIndex{23}$

$\dfrac{\left(a^{2}\right)^{3}}{a^{4}}$

Exercise $\PageIndex{24}$

$\dfrac{\left(p^{3}\right)^{4}}{p^{5}}$

Answer

$p^{7}$

Exercise $\PageIndex{25}$

$\dfrac{\left(y^{3}\right)^{4}}{y^{10}}$

Exercise $\PageIndex{26}$

$\dfrac{\left(x^{4}\right)^{5}}{x^{15}}$

Answer

$x^{5}$

Exercise $\PageIndex{27}$

$\dfrac{u^{6}}{\left(u^{3}\right)^{2}}$

Exercise $\PageIndex{28}$

$\dfrac{v^{20}}{\left(v^{4}\right)^{5}}$

Answer

1

Exercise $\PageIndex{29}$

$\dfrac{m^{12}}{\left(m^{8}\right)^{3}}$

Exercise $\PageIndex{30}$

$\dfrac{n^{8}}{\left(n^{6}\right)^{4}}$

Answer

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

Exercise $\PageIndex{31}$

$\left(\dfrac{p^{9}}{p^{3}}\right)^{5}$

Exercise $\PageIndex{32}$

$\left(\dfrac{q^{8}}{q^{2}}\right)^{3}$

Answer

$q^{18}$

Exercise $\PageIndex{33}$

$\left(\dfrac{r^{2}}{r^{6}}\right)^{3}$

Exercise $\PageIndex{34}$

$\left(\dfrac{m^{4}}{m^{7}}\right)^{4}$

Answer

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

Exercise $\PageIndex{35}$

$\left(\dfrac{p}{r^{11}}\right)^{2}$

Exercise $\PageIndex{36}$

$\left(\dfrac{a}{b^{6}}\right)^{3}$

Answer

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

Exercise $\PageIndex{37}$

$\left(\dfrac{w^{5}}{x^{3}}\right)^{8}$

Exercise $\PageIndex{38}$

$\left(\dfrac{y^{4}}{z^{10}}\right)^{5}$

Answer

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

Exercise $\PageIndex{39}$

$\left(\dfrac{2 j^{3}}{3 k}\right)^{4}$

Exercise $\PageIndex{40}$

$\left(\dfrac{3 m^{5}}{5 n}\right)^{3}$

Answer

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

Exercise $\PageIndex{41}$

$\left(\dfrac{3 c^{2}}{4 d^{6}}\right)^{3}$

Exercise $\PageIndex{42}$

$\left(\dfrac{5 u^{7}}{2 v^{3}}\right)^{4}$

Answer

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

Exercise $\PageIndex{43}$

$\left(\dfrac{k^{2} k^{8}}{k^{3}}\right)^{2}$

Exercise $\PageIndex{44}$

$\left(\dfrac{j^{2} j^{5}}{j^{4}}\right)^{3}$

Answer

$j^{9}$

Exercise $\PageIndex{45}$

$\dfrac{\left(t^{2}\right)^{5}\left(t^{4}\right)^{2}}{\left(t^{3}\right)^{7}}$

Exercise $\PageIndex{46}$

$\dfrac{\left(q^{3}\right)^{6}\left(q^{2}\right)^{3}}{\left(q^{4}\right)^{8}}$

Answer

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

Exercise $\PageIndex{47}$

$\dfrac{\left(-2 p^{2}\right)^{4}\left(3 p^{4}\right)^{2}}{\left(-6 p^{3}\right)^{2}}$

Exercise $\PageIndex{48}$

$\dfrac{\left(-2 k^{3}\right)^{2}\left(6 k^{2}\right)^{4}}{\left(9 k^{4}\right)^{2}}$

Answer

64$k^{6}$

Exercise $\PageIndex{49}$

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

Exercise $\PageIndex{50}$

$\dfrac{\left(-10 n^{2}\right)^{3}\left(4 n^{5}\right)^{2}}{\left(2 n^{8}\right)^{2}}$

Answer

−4,000

Divide Monomials

In the following exercises, divide the monomials.

Exercise $\PageIndex{51}$

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

Exercise $\PageIndex{52}$

63 $\nu^{10} \div 9 v^{2}$

Answer

7$v^{8}$

Exercise $\PageIndex{53}$

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

Exercise $\PageIndex{54}$

$-72 u^{12} \div 12 u^{4}$

Answer

$-6 u^{8}$

Exercise $\PageIndex{55}$

$\dfrac{45 a^{6} b^{8}}{-15 a^{10} b^{2}}$

Exercise $\PageIndex{56}$

$\dfrac{54 x^{9} y^{3}}{-18 x^{6} y^{15}}$

Answer

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

Exercise $\PageIndex{57}$

$\dfrac{15 r^{4} s^{9}}{18 r^{9} s^{2}}$

Exercise $\PageIndex{58}$

$\dfrac{20 m^{8} n^{4}}{30 m^{5} n^{9}}$

Answer

$\dfrac{2 m^{3}}{3 n^{5}}$

Exercise $\PageIndex{59}$

$\dfrac{18 a^{4} b^{8}}{-27 a^{9} b^{5}}$

Exercise $\PageIndex{60}$

$\dfrac{45 x^{5} y^{9}}{-60 x^{8} y^{6}}$

Answer

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

Exercise $\PageIndex{61}$

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

Exercise $\PageIndex{62}$

$\dfrac{65 a^{10} b^{8} c^{5}}{42 a^{7} b^{6} c^{8}}$

Answer

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

Exercise $\PageIndex{63}$

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

Exercise $\PageIndex{64}$

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

Answer

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

Exercise $\PageIndex{65}$

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

Exercise $\PageIndex{66}$

$\dfrac{\left(4 u^{2} v^{5}\right)\left(15 u^{3} v\right)}{\left(12 u^{3} v\right)\left(u^{4} v\right)}$

Answer

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

Mixed Practice

Exercise $\PageIndex{67}$

1. $24 a^{5}+2 a^{5}$
2. $24 a^{5}-2 a^{5}$
3. 24 $a^{5} \cdot 2 a^{5}$
4. 24 $a^{5} \div 2 a^{5}$

Exercise $\PageIndex{68}$

1. $15 n^{10}+3 n^{10}$
2. $15 n^{10}-3 n^{10}$
3. 15 $n^{10} \cdot 3 n^{10}$
4. 15 $n^{10} \div 3 n^{10}$

Answer

1. 18$n^{10}$
2. 12$n^{10}$
3. 45$n^{20}$
4. 5

Exercise $\PageIndex{69}$

1. $p^{4} \cdot p^{6}$
2. $\left(p^{4}\right)^{6}$

Exercise $\PageIndex{70}$

1. $q^{5} \cdot q^{3}$
2. $\left(q^{5}\right)^{3}$

Answer

1. $q^{8}$
2. $q^{15}$

Exercise $\PageIndex{71}$

1. $\dfrac{y^{3}}{y}$
2. $\dfrac{y}{y^{3}}$

Exercise $\PageIndex{72}$

1. $\dfrac{z^{6}}{z^{5}}$
2. $\dfrac{z^{5}}{z^{6}}$

Answer

1. z
2. $\dfrac{1}{z}$

Exercise $\PageIndex{73}$

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

Exercise $\PageIndex{74}$

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

Answer

6$y^{6}$

Exercise $\PageIndex{75}$

$\dfrac{27 a^{7}}{3 a^{3}}+\dfrac{54 a^{9}}{9 a^{5}}$

Exercise $\PageIndex{76}$

$\dfrac{32 c^{11}}{4 c^{5}}+\dfrac{42 c^{9}}{6 c^{3}}$

Answer

15$c^{6}$

Exercise $\PageIndex{77}$

$\dfrac{32 y^{5}}{8 y^{2}}-\dfrac{60 y^{10}}{5 y^{7}}$

Exercise $\PageIndex{78}$

$\dfrac{48 x^{6}}{6 x^{4}}-\dfrac{35 x^{9}}{7 x^{7}}$

Answer

3$x^{2}$

Exercise $\PageIndex{79}$

$\dfrac{63 r^{6} s^{3}}{9 r^{4} s^{2}}-\dfrac{72 r^{2} s^{2}}{6 s}$

Exercise $\PageIndex{80}$

$\dfrac{56 y^{4} z^{5}}{7 y^{3} z^{3}}-\dfrac{45 y^{2} z^{2}}{5 y}$

Answer

$-y z^{2}$

Everyday Math

Exercise $\PageIndex{81}$

Memory One megabyte is approximately $10^6$ bytes. One gigabyte is approximately $10^9$ bytes. How many megabytes are in one gigabyte?

Exercise $\PageIndex{82}$

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

Answer

$10^{3}$

Writing Exercises

Exercise $\PageIndex{83}$

Jennifer thinks the quotient $\dfrac{a^{24}}{a^{6}}$ simplifies to $a^{4} .$ What is wrong with her reasoning?

Exercise $\PageIndex{84}$

Maurice simplifies the quotient $\dfrac{d^{7}}{d}$ by writing $\dfrac{\not{d}^7}{\not{d}}=7 .$ What is wrong with his reasoning?

Answer

Answers will vary.

Exercise $\PageIndex{85}$

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

Exercise $\PageIndex{86}$

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

Answer

Answers will vary.

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?