Dividing Monomials

Example 6.72

Find the quotient: \(56 x^{7} \div 8 x^{3}\).

Solution

	\(56 x^{7} \div 8 x^{3}\)
Rewrite as a fraction.	\(\dfrac{56 x^{7}}{8 x^{3}}\)
Use fraction multiplication.	\(\dfrac{56}{8} \cdot \dfrac{x^{7}}{x^{3}}\)
Simplify and use the Quotient Property.	\(7 x^{4}\)

Try It 6.143

Find the quotient: \(42 y^{9} \div 6 y^{3}\).

Try It 6.144

Find the quotient: \(48 z^{8} \div 8 z^{2}\).

Example 6.73

Find the quotient: \(\dfrac{45 a^{2} b^{3}}{-5 a b^{5}}\).

Solution

	\(\dfrac{45 a^{2} b^{3}}{-5 a b^{5}}\)
Use fraction multiplication.	\(\dfrac{45}{-5} \cdot \dfrac{a^{2}}{a} \cdot \dfrac{b^{3}}{b^{5}}\)
Simplify and use the Quotient Property.	\(-9 \cdot a \cdot \dfrac{1}{b^{2}}\)
Multiply.	\(-\dfrac{9 a}{b^{2}}\)

Try It 6.145

Find the quotient: \(\dfrac{-72 a^{7} b^{3}}{8 a^{12} b^{4}}\).

Try It 6.146

Find the quotient: \(\dfrac{-63 c^{8} d^{3}}{7 c^{12} d^{2}}\).

Example 6.74

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

Solution

	\(\dfrac{24 a^{5} b^{3}}{48 a b^{4}}\)
Use fraction multiplication.	\(\dfrac{24}{48} \cdot \dfrac{a^{5}}{a} \cdot \dfrac{b^{3}}{b^{4}}\)
Simplify and use the Quotient Property.	\( \dfrac{1}{2} \cdot a^{4} \cdot \dfrac{1}{b} \)
Multiply.	\(\dfrac{a^{4}}{2 b}\)

Try It 6.147

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

Try It 6.148

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

Example 6.75

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

Solution

Be very careful to simplify 1421 by dividing out a common factor, and to simplify the variables by subtracting their exponents.

	\(\dfrac{14 x^{7} y^{12}}{21 x^{11} y^{6}}\)
Simplify and use the Quotient Property.	\(\dfrac{2 y^{6}}{3 x^{4}}\)

Try It 6.149

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

Try It 6.150

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

Example 6.76

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

Solution

	\(\dfrac{\left(6 x^{2} y^{3}\right)\left(5 x^{3} y^{2}\right)}{\left(3 x^{4} y^{5}\right)}\)
Simplify the numerator.	\(\dfrac{30 x^{5} y^{5}}{3 x^{4} y^{5}}\)
Simplify.	\(10x\)

Try It 6.151

Find the quotient: \(\dfrac{\left(6 a^{4} b^{5}\right)\left(4 a^{2} b^{5}\right)}{12 a^{5} b^{8}}\).

Try It 6.152

Find the quotient: \(\dfrac{\left(-12 x^{6} y^{9}\right)\left(-4 x^{5} y^{8}\right)}{-12 x^{10} y^{12}}\).