10.4 Divide monomials (Page 3/5)

Prealgebra

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\begin{matrix} {(\frac{2}{3})}^{3} & \overset{?}{=} & \frac{2^{3}}{3^{3}} \\ \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} & \overset{?}{=} & \frac{8}{27} \\ \frac{8}{27} & = & \frac{8}{27} ✓ \end{matrix}

Simplify:

ⓐ ${(\frac{5}{8})}^{2}$
ⓑ ${(\frac{x}{3})}^{4}$
ⓒ ${(\frac{y}{m})}^{3}$

Solution

ⓐ

Use the Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$ .
Simplify.

ⓑ

Use the Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$ .
Simplify.

ⓒ

Raise the numerator and denominator to the third power.

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Simplify:

ⓐ ${(\frac{7}{9})}^{2}$
ⓑ ${(\frac{y}{8})}^{3}$
ⓒ ${(\frac{p}{q})}^{6}$

ⓐ $\frac{49}{81}$
ⓑ $\frac{y^{3}}{512}$
ⓒ $\frac{p^{6}}{q^{6}}$

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Simplify:

ⓐ ${(\frac{1}{8})}^{2}$
ⓑ ${(\frac{−5}{m})}^{3}$
ⓒ ${(\frac{r}{s})}^{4}$

ⓐ $\frac{1}{64}$
ⓑ $- \frac{125}{m^{3}}$
ⓒ $\frac{r^{4}}{s^{4}}$

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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

\begin{matrix} Product Property & a^{m} \cdot a^{n} = a^{m + n} \\ Power Property & {(a^{m})}^{n} = a^{m \cdot n} \\ Product to a Power Property & {(a b)}^{m} = a^{m} b^{m} \\ Quotient Property & \frac{a^{m}}{a^{n}} = a^{m - n}, a \neq 0, m > n \\ \frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}, a \neq 0, n > m \\ Zero Exponent Definition & a^{0} = 1, a \neq 0 \\ Quotient to a Power Property & {(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, b \neq 0 \end{matrix}

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

Solution

	$\frac{{(x^{2})}^{3}}{x^{5}}$
Multiply the exponents in the numerator, using the Power Property.	$\frac{x^{6}}{x^{5}}$
Subtract the exponents.	$x$

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

a ¹¹

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

b ¹⁹

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

Solution

	$\frac{m^{8}}{{(m^{2})}^{4}}$
Multiply the exponents in the numerator, using the Power Property.	$\frac{m^{8}}{m^{8}}$
Subtract the exponents.	$m^{0}$

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

k ²

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

$\frac{1}{d}$

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Simplify: ${(\frac{x^{7}}{x^{3}})}^{2} .$

Solution

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

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Simplify: ${(\frac{f^{14}}{f^{8}})}^{2} .$

f ¹²

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Simplify: ${(\frac{b^{6}}{b^{11}})}^{2} .$

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

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Simplify: ${(\frac{p^{2}}{q^{5}})}^{3} .$

Solution

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

	${(\frac{p^{2}}{q^{5}})}^{3}$
Raise the numerator and denominator to the third power using the Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$	$\frac{{(p^{2})}^{3}}{{(q^{5})}^{3}}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$	$\frac{p^{6}}{q^{15}}$

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Simplify: ${(\frac{m^{3}}{n^{8}})}^{5} .$

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

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Simplify: ${(\frac{t^{10}}{u^{7}})}^{2} .$

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

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Simplify: ${(\frac{2 x^{3}}{3 y})}^{4} .$

Solution

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

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

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

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

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

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

Solution

	$\frac{{(y^{2})}^{3} {(y^{2})}^{4}}{{(y^{5})}^{4}}$
Use the Power Property.	$\frac{(y^{6}) (y^{8})}{y^{20}}$
Add the exponents in the numerator, using the Product Property.	$\frac{y^{14}}{y^{20}}$
Use the Quotient Property.	$\frac{1}{y^{6}}$

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

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

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

9 x ⁵

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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.

Find the quotient: $56 x^{5} \div 7 x^{2} .$

Solution

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

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Find the quotient: $63 x^{8} \div 9 x^{4} .$

7 x ⁴

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Find the quotient: $96 y^{11} \div 6 y^{8} .$

16 y ³

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

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

Solution

	$\frac{42 x^{2} y^{3}}{−7 x y^{5}}$
Use fraction multiplication.	$\frac{42}{−7} \cdot \frac{x^{2}}{x} \cdot \frac{y^{3}}{y^{5}}$
Simplify and use the Quotient Property.	$−6 \cdot x \cdot \frac{1}{y^{2}}$
Multiply.	$- \frac{6 x}{y^{2}}$

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

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

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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