Dividing Monomials

Summary of Exponent Properties

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

Product Property	\(a^{m} \cdot a^{n}=a^{m+n}\)m+n
Power Property	\(\left(a^{m}\right)^{n}=a^{m \cdot n}\)
Product to a Power	\((a b)^{m}=a^{m} b^{m}\)
Quotient Property	\( \begin{array}{l} \dfrac{a^{m}}{b^{m}}=a^{m-n}, a \neq 0, m>n \\ \dfrac{a^{m}}{a^{n}}=\dfrac{1}{a^{n-m}}, a \neq 0, n > m \end{array} \)
Zero Exponent Definition	\(a^{0}=1, a \neq 0\)
Quotient to a Power Property	\(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}, b \neq 0\)

Example 6.65

Simplify: \(\dfrac{\left(y^{4}\right)^{2}}{y^{6}}\).

Solution

	\(\dfrac{\left(y^{4}\right)^{2}}{y^{6}}\)
Multiply the exponents in the numerator.	\(\dfrac{y^{8}}{y^{6}}\)
Subtract the exponents.	\(y^{2}\)

Try It 6.129

Simplify: \(\dfrac{\left(m^{5}\right)^{4}}{m^{7}}\).

Try It 6.130

Simplify: \(\dfrac{\left(k^{2}\right)^{6}}{k^{7}}\).

Example 6.66

Simplify: \(\dfrac{b^{12}}{\left(b^{2}\right)^{6}}\).

Solution

	\(\dfrac{b^{12}}{\left(b^{2}\right)^{6}}\)
Multiply the exponents in the numerator.	\(\dfrac{b^{12}}{b^{12}}\)
Subtract the exponents.	\(b^{0}\)
Simplify.	\(1\)

Try It 6.131

Simplify: \(\dfrac{n^{12}}{\left(n^{3}\right)^{4}}\).

Try It 6.132

Simplify: \(\dfrac{x^{15}}{\left(x^{3}\right)^{5}}\).

Example 6.67

Simplify: \(\left(\dfrac{y^{9}}{y^{4}}\right)^{2}\).

Solution

	\(\left(\dfrac{y^{9}}{y^{4}}\right)^{2}\)
Remember parentheses come before exponents. Notice the bases are the same, so we can simplify inside the parentheses. Subtract the exponents.	\(\left(y^{5}\right)^{2}\)
Multiply the exponents.	\(y^{10}\)

Try It 6.133

Simplify: \(\left(\dfrac{r^{5}}{r^{3}}\right)^{4}\).

Try It 6.134

Simplify: \(\left(\dfrac{v^{6}}{v^{4}}\right)^{3}\).

Example 6.68

Simplify: \(\left(\dfrac{j^{2}}{k^{3}}\right)^{4}\).

Solution

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

	\(\left(\dfrac{j^{2}}{k^{3}}\right)^{4}\)
Raise the numerator and denominator to the third power using the Quotient to a Power Property, \(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}} .\).
Use the Power Property and simplify.	\(\dfrac{j^{8}}{k^{12}}\)

Try It 6.135

Simplify: \(\left(\dfrac{a^{3}}{b^{2}}\right)^{4}\).

Try It 6.136

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

Example 6.69

Simplify: \(\left(\dfrac{2 m^{2}}{5 n}\right)^{4}\).

Solution

	\(\left(\dfrac{2 m^{2}}{5 n}\right)^{4}\)
Raise the numerator and denominator to the fourth power, using the Quotient to a Power Property, \(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}\).	\(\dfrac{\left(2 m^{2}\right)^{4}}{(5 n)^{4}}\))4
Raise each factor to the fourth power.	\(\dfrac{\left(2 m^{2}\right)^{4}}{(5 n)^{4}}\)
Use the Power Property and simplify.	\(\dfrac{16 m^{8}}{625 n^{4}}\)

Try It 6.137

Simplify: \(\left(\dfrac{7 x^{3}}{9 y}\right)^{2}\).

Try It 6.138

Simplify: \(\left(\dfrac{3 x^{4}}{7 y}\right)^{2}\).

Example 6.70

Simplify: \(\dfrac{\left(x^{3}\right)^{4}\left(x^{2}\right)^{5}}{\left(x^{6}\right)^{5}}\).

Solution

	\(\dfrac{\left(x^{3}\right)^{4}\left(x^{2}\right)^{5}}{\left(x^{6}\right)^{5}}\)
Use the Power Property, \(\left(a^{m}\right)^{n}=a^{m \cdot n}\).	\(\dfrac{\left(x^{12}\right)\left(x^{10}\right)}{\left(x^{30}\right)}\)
Add the exponents in the numerator.	\(\dfrac{x^{22}}{x^{30}}\)
Use the Quotient Property, aman=1an−m.	\(\dfrac{1}{x^{8}}\)

Try It 6.139

Simplify: \(\dfrac{\left(a^{2}\right)^{3}\left(a^{2}\right)^{4}}{\left(a^{4}\right)^{5}}\).

Try It 6.140

Simplify:\(\dfrac{\left(p^{3}\right)^{4}\left(p^{5}\right)^{3}}{\left(p^{7}\right)^{6}}\).

Example 6.71

Simplify: \(\dfrac{\left(10 p^{3}\right)^{2}}{(5 p)^{3}\left(2 p^{5}\right)^{4}}\).

Solution

	\(\dfrac{\left(10 p^{3}\right)^{2}}{(5 p)^{3}\left(2 p^{5}\right)^{4}}\)
Use the Product to a Power Property, \((a b)^{m}=a^{m} b^{m}\).	\(\dfrac{(10)^{2}\left(p^{3}\right)^{2}}{(5)^{3}(p)^{3}(2)^{4}\left(p^{5}\right)^{4}}\)
Use the Power Property, \(\left(a^{m}\right)^{n}=a^{m \cdot n}\).	\(\dfrac{100 p^{6}}{125 p^{3} \cdot 16 p^{20}}\)
Add the exponents in the denominator.	\(\dfrac{100 p^{6}}{125 \cdot 16 p^{23}}\)
Use the Quotient Property, \(\dfrac{a^{m}}{a^{n}}=\dfrac{1}{a^{n-m}}\).	\(\dfrac{100}{125 \cdot 16 p^{17}}\)
Simplify.	\(\dfrac{1}{20 p^{17}}\)

Try It 6.141

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

Try It 6.142

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