
# 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. $$\dfrac{7^{16}}{7^{2}}$$
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. $$\dfrac{9^{15}}{9^{5}}$$
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. $$\dfrac{8^{3}}{8^{5}}$$
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. $$\dfrac{10}{10^{3}}$$
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. $$k^{0}$$
1. 1
2. 1

Exercise $$\PageIndex{11}$$

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

Exercise $$\PageIndex{12}$$

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

Exercise $$\PageIndex{13}$$

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

Exercise $$\PageIndex{14}$$

1. $$(6 y)^{0}$$
2. $$6 y^{0}$$
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. $$\left(-93 c^{7} d^{15}\right)^{0}$$
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. $$(15 r)^{0}-(22 s)^{0}$$
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. $$\left(\dfrac{a}{b}\right)^{5}$$
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. $$\left(\dfrac{10}{3 q}\right)^{4}$$
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 $$\PageIndex{24}$$

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

$$p^{7}$$

Exercise $$\PageIndex{25}$$

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

Exercise $$\PageIndex{26}$$

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

$$x^{5}$$

Exercise $$\PageIndex{27}$$

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

Exercise $$\PageIndex{28}$$

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

1

Exercise $$\PageIndex{29}$$

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

Exercise $$\PageIndex{30}$$

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

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

Exercise $$\PageIndex{31}$$

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

Exercise $$\PageIndex{32}$$

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

$$q^{18}$$

Exercise $$\PageIndex{33}$$

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

Exercise $$\PageIndex{34}$$

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

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

Exercise $$\PageIndex{35}$$

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

Exercise $$\PageIndex{36}$$

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

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

Exercise $$\PageIndex{37}$$

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

Exercise $$\PageIndex{38}$$

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

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

Exercise $$\PageIndex{39}$$

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

Exercise $$\PageIndex{40}$$

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

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

Exercise $$\PageIndex{41}$$

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

Exercise $$\PageIndex{42}$$

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

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

Exercise $$\PageIndex{43}$$

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

Exercise $$\PageIndex{44}$$

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

$$j^{9}$$

Exercise $$\PageIndex{45}$$

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

Exercise $$\PageIndex{46}$$

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

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

Exercise $$\PageIndex{47}$$

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

Exercise $$\PageIndex{48}$$

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

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 $$\PageIndex{50}$$

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

−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$$\nu^{10} \div 9 v^{2}$$

7$$v^{8}$$

Exercise $$\PageIndex{53}$$

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

Exercise $$\PageIndex{54}$$

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

$$-6 u^{8}$$

Exercise $$\PageIndex{55}$$

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

Exercise $$\PageIndex{56}$$

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

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

Exercise $$\PageIndex{57}$$

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

Exercise $$\PageIndex{58}$$

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

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

Exercise $$\PageIndex{59}$$

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

Exercise $$\PageIndex{60}$$

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

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

Exercise $$\PageIndex{61}$$

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

Exercise $$\PageIndex{62}$$

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

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

Exercise $$\PageIndex{63}$$

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

Exercise $$\PageIndex{64}$$

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

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

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 $$\PageIndex{66}$$

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

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

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$$n^{10} \div 3 n^{10}$$
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. $$\left(q^{5}\right)^{3}$$
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. $$\dfrac{z^{5}}{z^{6}}$$
1. z
2. $$\dfrac{1}{z}$$

Exercise $$\PageIndex{73}$$

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

Exercise $$\PageIndex{74}$$

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

6$$y^{6}$$

Exercise $$\PageIndex{75}$$

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

Exercise $$\PageIndex{76}$$

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

15$$c^{6}$$

Exercise $$\PageIndex{77}$$

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

Exercise $$\PageIndex{78}$$

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

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 $$\PageIndex{80}$$

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

$$-y z^{2}$$

## 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?

$$10^{3}$$

## 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?

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

Exercise $$\PageIndex{86}$$

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