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# 10.7: Divide Monomials (Part 2)

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## Simplify Expressions by Applying Several Properties

We'll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

Summary of Exponent Properties

If a, b are real numbers and m, n are whole numbers, then

 Product Property $$a^m • a^n = a^{m + n}$$ Power Property $$(a^m)^n = a^{m • n}$$ Product to a Power Property $$(ab)^m = a^mb^m$$ Quotient Property $$\dfrac{a^{m}}{a^{n}} = a^{m − n},\quad a ≠ 0,\, m > n$$ $$\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n-m}}, \quad a ≠ 0, \,n > m$$ Zero Exponent Property $$a^0 = 1, \quad a ≠ 0$$ Quotient to a Power Property $$\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}, b ≠ 0$$

Example $$\PageIndex{8}$$:

Simplify: $$\dfrac{(x^{2})^{3}}{x^{5}}$$.

Solution

 Multiply the exponents in the numerator, using the Power Property. $$\dfrac{x^{6}}{x^{5}} \label{10.4.46}$$ Subtract the exponents. $$x \label{10.4.47}$$

Exercise $$\PageIndex{15}$$:

Simplify: $$\dfrac{(a^{4})^{5}}{a^{9}}$$.

$$a^{11}$$

Exercise $$\PageIndex{16}$$:

Simplify: $$\dfrac{(b^{5})^{6}}{b^{11}}$$.

$$b^{19}$$

Example $$\PageIndex{9}$$:

Simplify: $$\dfrac{(m^{8})}{(m^{2})^{4}}$$.

Solution

 Multiply the exponents in the numerator, using the Power Property. $$\dfrac{m^{8}}{m^{8}} \label{10.4.48}$$ Subtract the exponents. $$m^{0} \label{10.4.49}$$ Zero power property 1

Exercise $$\PageIndex{17}$$:

Simplify: $$\dfrac{(k^{11}}{(k^{3})^{3}}$$.

$$k^2$$

Exercise $$\PageIndex{18}$$:

Simplify: $$\dfrac{(d^{23}}{(d^{4})^{6}}$$.

$$\frac{1}{d}$$

Example $$\PageIndex{10}$$:

Simplify: $$\left(\dfrac{x^{7}}{x^{3}}\right)^{2}$$.

Solution

 Remember parentheses come before exponents, and the bases are the same so we can simplify inside the parentheses. Subtract the exponents. $$(x^{7-3})^{2} \label{10.4.50}$$ Simplify. $$(x^{4})^{2} \label{10.4.51}$$ Multiply the exponents. $$x^{8} \label{10.4.52}$$

Exercise $$\PageIndex{19}$$:

Simplify: $$\left(\dfrac{f^{14}}{f^{8}}\right)^{2}$$.

$$f^{12}$$

Exercise $$\PageIndex{20}$$:

Simplify: $$\left(\dfrac{b^{6}}{b^{11}}\right)^{2}$$.

$$\frac{1}{b^{10}}$$

Example $$\PageIndex{11}$$:

Simplify: $$\left(\dfrac{p^{2}}{q^{5}}\right)^{3}$$.

Solution

Here we cannot simplify inside the parentheses first, since the bases are not the same.

 Raise the numerator and denominator to the third power using the Quotient to a Power Property, $$\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}$$ $$\dfrac{(p^{2})^{3}}{(q^{5})^{3}} \label{10.4.53}$$ Use the Power Property, (am)n = am • n. $$\dfrac{p^{6}}{q^{15}} \label{10.4.54}$$

Exercise $$\PageIndex{21}$$:

Simplify: $$\left(\dfrac{m^{3}}{n^{8}}\right)^{5}$$.

$$\frac{m^{15}}{n^{40}}$$

Exercise $$\PageIndex{22}$$:

Simplify: $$\left(\dfrac{t^{10}}{u^{7}}\right)^{2}$$.

$$\frac{t^{20}}{u^{14}}$$

Example $$\PageIndex{12}$$:

Simplify: $$\left(\dfrac{2x^{3}}{3y}\right)^{4}$$.

Solution

 Raise the numerator and denominator to the fourth power using the Quotient to a Power Property. $$\dfrac{(2x^{3})^{4}}{(3y)^{4}} \label{10.4.55}$$ Raise each factor to the fourth power, using the Power to a Power Property. $$\dfrac{2^{4} (x^{3})^{4}}{3^{4} y^{4}} \label{10.4.56}$$ Use the Power Property and simplify. $$\dfrac{16x^{12}}{81y^{4}} \label{10.4.57}$$

Exercise $$\PageIndex{23}$$:

Simplify: $$\left(\dfrac{5b}{9c^{3}}\right)^{2}$$.

$$\frac{25b^2}{81c^6}$$

Exercise $$\PageIndex{24}$$:

Simplify: $$\left(\dfrac{4p^{4}}{7q^{5}}\right)^{3}$$.

$$\frac{64p^{12}}{343q^{15}}$$

Example $$\PageIndex{13}$$:

Simplify: $$\dfrac{(y^{2})^{3} (y^{2})^{4}}{(y^{5})^{4}}$$.

Solution

 Use the Power Property. $$\dfrac{(y^{6})(y^{8})}{y^{20}} \label{10.4.58}$$ Add the exponents in the numerator, using the Product Property. $$\dfrac{y^{14}}{y^{20}} \label{10.4.59}$$ Use the Quotient Property. $$\dfrac{1}{y^{6}} \label{10.4.60}$$

Exercise $$\PageIndex{25}$$

Simplify: $$\dfrac{(y^{4})^{4} (y^{3})^{5}}{(y^{7})^{6}}$$.

$$\frac{1}{y^{11}}$$

Exercise $$\PageIndex{26}$$

Simplify: $$\dfrac{(3x^{4})^{2} (x^{3})^{4}}{(x^{5})^{3}}$$.

$$9x^5$$

## Divide Monomials

We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials.

Example $$\PageIndex{14}$$:

Find the quotient: 56x5 ÷ 7x2.

Solution

 Rewrite as a fraction. $$\dfrac{56x^{5}}{7x^{2}} \label{10.4.61}$$ Use fraction multiplication to separate the number part from the variable part. $$\dfrac{56}{7} \cdot \dfrac{x^{5}}{x^{2}} \label{10.4.62}$$ Use the Quotient Property. $$8x^{3} \label{10.4.63}$$

Exercise $$\PageIndex{27}$$:

Find the quotient: 63x8 ÷ 9x4.

$$7x^4$$

Exercise $$\PageIndex{28}$$:

Find the quotient: 96y11 ÷ 6y8.

$$16y^3$$

When we divide monomials with more than one variable, we write one fraction for each variable.

Example $$\PageIndex{15}$$:

Find the quotient: $$\dfrac{42x^{2} y^{3}}{−7xy^{5}}$$.

Solution

 Use fraction multiplication. $$\dfrac{42}{-7} \cdot \dfrac{x^{2}}{x} \cdot \dfrac{y^{3}}{y^{5}} \label{10.4.64}$$ Simplify and use the Quotient Property. $$-6 \cdot x \cdot \dfrac{1}{y^{2}} \label{10.4.65}$$ Multiply. $$- \dfrac{6x}{y^{2}} \label{10.4.66}$$

Exercise $$\PageIndex{29}$$:

Find the quotient: $$\dfrac{-84x^{8} y^{3}}{7x^{10} y^{2}}$$.

$$-\frac{12y}{x^2}$$

Exercise $$\PageIndex{30}$$:

Find the quotient: $$\dfrac{-72a^{4} b^{5}}{−8a^{9} b^{5}}$$.

$$\frac{9}{a^5}$$

Example $$\PageIndex{16}$$:

Find the quotient: $$\dfrac{24a^{5} b^{3}}{48ab^{4}}$$.

Solution

 Use fraction multiplication. $$\dfrac{24}{48} \cdot \dfrac{a^{5}}{a} \cdot \dfrac{b^{3}}{b^{4}} \label{10.4.67}$$ Simplify and use the Quotient Property. $$\dfrac{1}{2} \cdot a^{4} \cdot \dfrac{1}{b} \label{10.4.68}$$ Multiply. $$\dfrac{a^{4}}{2b} \label{10.4.69}$$

Exercise $$\PageIndex{31}$$:

Find the quotient: $$\dfrac{16a^{7} b^{6}}{24ab^{8}}$$.

$$\frac{2a^6}{3b^2}$$

Exercise $$\PageIndex{32}$$:

Find the quotient: $$\dfrac{27p^{4} q^{7}}{-45p^{12} q}$$.

$$-\frac{3q^6}{5p^8}$$

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

Example $$\PageIndex{17}$$:

Find the quotient: $$\dfrac{14x^{7} y^{12}}{21x^{11} y^{6}}$$.

Solution

 Simplify and use the Quotient Property. $$\dfrac{2y^{6}}{3x^{4}} \label{10.4.70}$$

Be very careful to simplify $$\dfrac{14}{21}$$ by dividing out a common factor, and to simplify the variables by subtracting their exponents.

Exercise $$\PageIndex{33}$$:

Find the quotient: $$\dfrac{28x^{5} y^{14}}{49x^{9} y^{12}}$$.

$$\frac{4y^2}{7x^4}$$

Exercise $$\PageIndex{34}$$:

Find the quotient: $$\dfrac{30m^{5} n^{11}}{48m^{10} n^{14}}$$.

$$\frac{5}{8m^5 n^3}$$

In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we'll first find the product of two monomials in the numerator before we simplify the fraction.

Example $$\PageIndex{18}$$:

Find the quotient: $$\dfrac{(3x^{3} y^{2})(10x^{2} y^{3})}{6x^{4} y^{5}}$$.

Solution

Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.

 Simplify the numerator. $$\dfrac{30x^{5} y^{5}}{6x^{4} y^{5}} \label{10.4.71}$$ Simplify, using the Quotient Rule. $$5x \label{10.4.72}$$

Exercise $$\PageIndex{35}$$:

Find the quotient: $$\dfrac{(3x^{4} y^{5})(8x^{2} y^{5})}{12x^{5} y^{8}}$$.

$$2xy^2$$

Exercise $$\PageIndex{36}$$:

Find the quotient: $$\dfrac{(-6a^{6} b^{9})(-8a^{5} b^{8})}{-12a^{10} b^{12}}$$.

$$-4ab^5$$

## Practice Makes Perfect

### Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

1. $$\dfrac{4^{8}}{4^{2}}$$
2. $$\dfrac{3^{12}}{3^{4}}$$
3. $$\dfrac{x^{12}}{x^{3}}$$
4. $$\dfrac{u^{9}}{u^{3}}$$
5. $$\dfrac{r^{5}}{r}$$
6. $$\dfrac{y^{4}}{y}$$
7. $$\dfrac{y^{4}}{y^{20}}$$
8. $$\dfrac{x^{10}}{x^{30}}$$
9. $$\dfrac{10^{3}}{10^{15}}$$
10. $$\dfrac{r^{2}}{r^{8}}$$
11. $$\dfrac{a}{a^{9}}$$
12. $$\dfrac{2}{2^{5}}$$

### Simplify Expressions with Zero Exponents

In the following exercises, simplify.

1. 50
2. 100
3. a0
4. x0
5. −70
6. −40
7. (a) (10p)0 (b) 10p0
8. (a) (3a)0 (b) 3a0
9. (a) (−27x5y)0 (b) −27x5y0
10. (a) (−92y8z)0 (b) −92y8z0
11. (a) 150 (b) 151
12. (a) −60 (b) −61
13. 2 • x0 + 5 • y0
14. 8 • m0 − 4 • n0

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

In the following exercises, simplify.

1. $$\left(\dfrac{3}{2}\right)^{5}$$
2. $$\left(\dfrac{4}{5}\right)^{3}$$
3. $$\left(\dfrac{m}{6}\right)^{3}$$
4. $$\left(\dfrac{p}{2}\right)^{5}$$
5. $$\left(\dfrac{x}{y}\right)^{10}$$
6. $$\left(\dfrac{a}{b}\right)^{8}$$
7. $$\left(\dfrac{a}{3b}\right)^{2}$$
8. $$\left(\dfrac{2x}{y}\right)^{4}$$

### Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

1. $$\dfrac{(x^{2})^{4}}{x^{5}}$$
2. $$\dfrac{(y^{4})^{3}}{y^{7}}$$
3. $$\dfrac{(u^{3})^{4}}{u^{10}}$$
4. $$\dfrac{(y^{2})^{5}}{y^{6}}$$
5. $$\dfrac{y^{8}}{(y^{5})^{2}}$$
6. $$\dfrac{p^{11}}{(p^{5})^{3}}$$
7. $$\dfrac{r^{5}}{(r^{4} \cdot r}$$
8. $$\dfrac{a^{3} \cdot a^{4}}{(a^{7}}$$
9. $$\left(\dfrac{x^{2}}{x^{8}}\right)^{3}$$
10. $$\left(\dfrac{u}{u^{10}}\right)^{2}$$
11. $$\left(\dfrac{a^{4} \cdot a^{6}}{a^{3}}\right)^{2}$$
12. $$\left(\dfrac{x^{3 \cdot x^{8}}}{x^{4}}\right)^{3}$$
13. $$\dfrac{(y^{3})^{5}}{(y^{4})^{3}}$$
14. $$\dfrac{(z^{6})^{2}}{(z^{2})^{4}}$$
15. $$\dfrac{(x^{3})^{6}}{(x^{4})^{7}}$$
16. $$\dfrac{(x^{4})^{8}}{(x^{5})^{7}}$$
17. $$\left(\dfrac{2r^{3}}{5s}\right)^{4}$$
18. $$\left(\dfrac{3m^{2}}{4n}\right)^{3}$$
19. $$\left(\dfrac{3y^{2} \cdot y^{5}}{y^{15} \cdot y^{8}}\right)^{0}$$
20. $$\left(\dfrac{15z^{4} \cdot z^{9}}{0.3z^{2}}\right)^{0}$$
21. $$\dfrac{(r^{2})^{5} (r^{4})^{2}}{(r^{3})^{7}}$$
22. $$\dfrac{(p^{4})^{2} (p^{3})^{5}}{(p^{2})^{9}}$$
23. $$\dfrac{(3x^{4})^{3} (2x^{3})^{2}}{(6x^{5})^{2}}$$
24. $$\dfrac{(-2y^{3})^{4} (3y^{4})^{2}}{(-6y^{3})^{2}}$$

### Divide Monomials

In the following exercises, divide the monomials.

1. 48b8 ÷ 6b2
2. 42a14 ÷ 6a2
3. 36x3 ÷ (−2x9)
4. 20u8 ÷ (−4u6)
5. $$\dfrac{18x^{3}}{9x^{2}}$$
6. $$\dfrac{36y^{9}}{4y^{7}}$$
7. $$\dfrac{-35x^{7}}{-42x^{13}}$$
8. $$\dfrac{18x^{5}}{-27x^{9}}$$
9. $$\dfrac{18r^{5} s}{3r^{3} s^{9}}$$
10. $$\dfrac{24p^{7} q}{6p^{2} q^{5}}$$
11. $$\dfrac{8mn^{10}}{64mn^{4}}$$
12. $$\dfrac{10a^{4} b}{50a^{2} b^{6}}$$
13. $$\dfrac{-12x^{4} y^{9}}{15x^{6} y^{3}}$$
14. $$\dfrac{48x^{11} y^{9} z^{3}}{36x^{6} y^{8} z^{5}}$$
15. $$\dfrac{64x^{5} y^{9} z^{7}}{48x^{7} y^{12} z^{6}}$$
16. $$\dfrac{(10u^{2} v)(4u^{3} v^{6})}{5u^{9} v^{2}}$$
17. $$\dfrac{(6m^{2} n)(5m^{4} n^{3})}{3m^{10} n^{2}}$$
18. $$\dfrac{(6a^{4} b^{3})(4ab^{5})}{(12a^{8} b)(a^{3} b)}$$
19. $$\dfrac{(4u^{5} v^{4})(15u^{8} v)}{(12u^{3} v)(u^{6} v)}$$

## Mixed Practice

1. (a) 24a5 + 2a5 (b) 24a5 − 2a5 (c) 24a5 • 2a5 (d) 24a5 ÷ 2a5
2. (a) 15n10 + 3n10 (b) 15n10 − 3n10 (c) 15n10 • 3n10 (d) 15n10 ÷ 3n10
3. (a) p4 • p6 (b) (p4)6
4. (a) q5 • q3 (b) (q5)3
5. (a) $$\dfrac{ y^{3}}{y}$$ (b) $$\dfrac{y}{y^{3}}$$
6. (a) $$\dfrac{z^{6}}{z^{5}}$$ (b) $$\dfrac{z^{5}}{z^{6}}$$
7. (8x5)(9x) ÷ 6x3
8. (4y5)(12y7) ÷ 8y2
9. $$\dfrac{27a^{7}}{3a^{3}} + \dfrac{54a^{9}}{9a^{5}}$$
10. $$\dfrac{32c^{11}}{4c^{5}} + \dfrac{42c^{9}}{6c^{3}}$$
11. $$\dfrac{32y^{5}}{8y^{2}} - \dfrac{60y^{10}}{5y^{7}}$$
12. $$\dfrac{48x^{6}}{6x^{4}} - \dfrac{35x^{9}}{7x^{7}}$$
13. $$\dfrac{63r^{6} s^{3}}{9r^{4} s^{2}} - \dfrac{72r^{2} s^{2}}{6s}$$
14. $$\dfrac{56y^{4} z^{5}}{7y^{3} z^{3}} - \dfrac{45y^{2} z^{2}}{5y}$$

## Everyday Math

1. Memory One megabyte is approximately 106 bytes. One gigabyte is approximately 109 bytes. How many megabytes are in one gigabyte?
2. Memory One megabyte is approximately 106 bytes. One terabyte is approximately 1012 bytes. How many megabytes are in one terabyte?

## Writing Exercises

1. Vic thinks the quotient $$\dfrac{x^{20}}{x^{4}}$$ simplifies to x5. What is wrong with his reasoning?
2. Mai simplifies the quotient $$\dfrac{y^{3}}{y}$$ by writing $$\dfrac{y^{3}}{y}$$ = 3. What is wrong with her reasoning?
3. When Dimple simplified −30 and (−3)0 she got the same answer. Explain how using the Order of Operations correctly gives different answers.
4. Roxie thinks n0 simplifies to 0. What would you say to convince Roxie she is wrong?

## 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?