Dividing Monomials
| Site: | Saylor Academy |
| Course: | MA007: Algebra |
| Book: | Dividing Monomials |
| Printed by: | Guest user |
| Date: | Tuesday, 15 July 2025, 7:40 AM |
Description
buttons.Introduction
Source: OpenStax, https://openstax.org/books/elementary-algebra/pages/6-5-divide-monomials
This work is licensed under a Creative Commons Attribution 4.0 License.
Simplify Expressions Using the Quotient Property for Exponents
Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below.
Summary of Exponent Properties for Multiplication
If \(text{a and b}\) are real numbers, and \(\text{m and n}\) are whole numbers, then
| Product Property | \(a^{m} \cdot a^{n}=a^{m+n}\) |
| Power Property | \(\left(a^{m}\right)^{n}=a^{m \cdot n}\) |
| Product to a Power | \((a b)^{m}=a^{m} b^{m}\) |
Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions - which are also quotients.
Equivalent Fractions Property
If \(a, b\), and \(c\) are whole numbers where \(b \neq 0, c \neq 0\),
\(\text{then} \frac{a}{b}=\frac{a \cdot c}{b \cdot c} \, \text{and} \, \frac{a \cdot c}{b \cdot c}=\frac{a}{b}\)
As before, we'll try to discover a property by looking at some examples.
| Consider | \(\frac{x^{5}}{x^{2}}\) | and | \(\frac{x^{2}}{x^{3}}\) |
| What do they mean? | \(\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}\) | \(\frac{x \cdot x}{x \cdot x \cdot x}\) | |
| Use the Equivalent Fractions Property. | \(\frac{\not {x} \cdot \not{x} \cdot x \cdot x \cdot x}{\not \not{x} \cdot \not{x}}\) | \(\frac{\not {x} \cdot \not{x} \cdot 1 }{\not \not{x} \cdot \not{x} \cdot \not{x}}\) | |
| Simplify. | \(x^{3}\) | \(\frac{1}{x}\) |
Notice, in each case the bases were the same and we subtracted exponents.
When the larger exponent was in the numerator, we were left with factors in the numerator.
When the larger exponent was in the denominator, we were left with factors in the denominator - notice the numerator of \(1\).
We write:
\(\begin{array}{cc}\frac{x^{5}}{x^{2}} & \frac{x^{2}}{x^{3}} \\ x^{5-2} & \frac{1}{x^{3-2}} \\ x^{3} & \frac{1}{x}\end{array}\)
This leads to the Quotient Property for Exponents.
Quotient Property for Exponents
If \(a\) is a real number, \(a \neq 0\), and \(m\) and \(n\) are whole numbers, then
\(\frac{a^{m}}{a^{n}}=a^{m-n}, m>n\) and \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}, n>m\)
A couple of examples with numbers may help to verify this property.
\(\begin{aligned} \frac{3^{4}}{3^{2}} &=3^{4-2} & \frac{5^{2}}{5^{3}} &=\frac{1}{5^{3-2}} \\ \frac{81}{9} &=3^{2} & \frac{25}{125} &=\frac{1}{5^{1}} \\ 9 &=9 \text{✓} & \frac{1}{5} &=\frac{1}{5} \text{✓} \end{aligned}\)
Example 6.59
Simplify:
- \(\frac{x^{9}}{x^{7}}\)
- \(\frac{3^{10}}{3^{2}}\).
Solution
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
-
Since \( 9 > 7\), there are more factors of x in the numerator. \(\frac{x^{9}}{x^{2}}\) Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=a^{m-n}\). \(X^{9-7}\) Simplify. \(x^{2}\) -
Since \(10 > 2\), there are more factors of x in the numerator. \(\frac{3^{10}}{3^{2}}\) Use the Quotient Property, aman=am−n. \(3^{10-2}\) Simplify. \(3^{8}\)
Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
Try It 6.117
Simplify:
- \(\frac{x^{15}}{x^{10}}\)
- \(\frac{6^{14}}{6^{5}}\).
Try It 6.118
Simplify:
- \(\frac{y^{43}}{y^{37}}\)
- \(\frac{10^{15}}{10^{7}}\).
Example 6.60
Simplify:
- \(\frac{b^{8}}{b^{12}}\)
- \(\frac{7^{3}}{7^{5}}\).
Solution
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
-
Since \(12 > 8\), there are more factors of \(b\) in the denominator. \(\frac{b^{s}}{b^{12}}\) Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}\). \(\frac{1}{b^{12-8}}\) Simplify. \(\frac{1}{b^{4}}\) -
Since \(5 > 3\), there are more factors of \(3\) in the denominator. \(\frac{7^{3}}{7^{5}}\) Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}\). \(\frac{1}{7^{5-3}}\) Simplify. \(\frac{1}{7^{2}}\) Simplify. \(\frac{1}{49}\)
Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.
Try It 6.119
Simplify:
- \(\frac{x^{18}}{x^{22}}\)
- \(\frac{12^{15}}{12^{30}}\).
Try It 6.120
Simplify:
- \(\frac{m^{7}}{m^{15}}\)
- \(\frac{9^{8}}{9^{19}}\).
Notice the difference in the two previous examples:
- If we start with more factors in the numerator, we will end up with factors in the numerator.
- If we start with more factors in the denominator, we will end up with factors in the denominator.
The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.
Example 6.61
Simplify:
- \(\frac{a^{5}}{a^{9}}\)
- \(\frac{x^{11}}{x^{7}}\).
Solution
-
Is the exponent of a larger in the numerator or denominator? Since \(9 > 5\), there are more \(a\)'s in the denominator and so we will end up with factors in the denominator.
\(\frac{a^{5}}{a^{9}}\) Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}\). \(\frac{1}{a^{9-5}}\) Simplify. \(\frac{1}{a^{4}}\)
-
Notice there are more factors of \(x\) in the numerator, since \(11 > 7\). So we will end up with factors in the numerator.
\(\frac{x^{11}}{X^{7}}\) Use the Quotient Property, \(\frac{a^{m}}{a^{n}}=\frac{1}{a^{n-m}}\). \(x^{11-1}\) Simplify. \(x^{4}\)
Try It 6.121
Simplify:
- \(\frac{b^{19}}{b^{11}}\)
- \(\frac{z^{5}}{z^{11}}\).
Try It 6.122
Simplify:
- \(\frac{p^{9}}{p^{17}}\)
- \(\frac{w^{13}}{w^{9}}\).
Simplify Expressions with an Exponent of Zero
A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like \(\dfrac{a^{m}}{a^{m}}\). From your earlier work with fractions, you know that:
\(\dfrac{2}{2}=1 \qquad \dfrac{17}{17}=1 \qquad \dfrac{-43}{-43}=1 \)
In words, a number divided by itself is \(1\). So, \(\dfrac{x}{x}=1\), for any \(x(x \neq 0)\), since any number divided by itself is \(1\) .
The Quotient Property for Exponents shows us how to simplify \(\dfrac{a^{m}}{a^{n}}\) when \(m>n\) and when \(n<m\) by subtracting exponents.
What if \(m=n\)?
Consider \(\dfrac{8}{8}\), which we know is \(1\).
| \(\dfrac{8}{8}=1\) | |
| Write \(8\) as \(23\). | \(\dfrac{2^{3}}{2^{2}}=1\) |
| Subtract exponents. | \(\dfrac{2^{3-3}}=1\) |
| Simplify. | \({2^{0}}=1\) |
Now we will simplify \(\dfrac{a^{m}}{a^{m}}\) in two ways to lead us to the definition of the zero exponent. In general, for \(a \neq 0\):
| \(\dfrac{a^{m}}{a^{m}}\) | \(\dfrac{a^{m}}{a^{m}}\) |
| \(a^{m-m}\) | \(\dfrac{\overbrace{\not{a} \cdot \not{a} \cdot \, \ldots \, \cdot \not{a}}^{\text{m factors}}}{\underbrace{\not{a} \cdot \not{a} \cdot \, \ldots \, \cdot \not{a}}_{\text{m factors}}}\) |
| \(a^{0}\) | \(1\) |
We see \(\dfrac{a^{m}}{a^{m}}\) simplifies to \(a^{0}\) and to \(1 .\) So \(a^{0}=1\).
Zero Exponent
If \(a\) is a non-zero number, then \(a^{0}=1\).
Any nonzero number raised to the zero power is \(1\).
In this text, we assume any variable that we raise to the zero power is not zero.
Example 6.62
Simplify:
- \(9^{0}\)
- \(n^{0}\).
Solution
The definition says any non-zero number raised to the zero power is \(1\).
|
|
\(9^{0}\) \(1\) |
|
|
\(n^{0}\) \(1\) |
Try It 6.123
Simplify:
- \(15^{0}\)
- \(m^{0}\).
Try It 6.124
Simplify:
- \(k^{0}\)
- \(29^{0}\).
Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.
What about raising an expression to the zero power? Let's look at \((2 x)^{0}\). We can use the product to a power rule to rewrite this expression.
| \((2 x)^{0}\) | |
| Use the product to a power rule. | \(2^{0} x^{0}\) |
| Use the zero exponent property. | \(1 \cdot 1\) |
| Simplify. | \(1\) |
Table 6.1
This tells us that any nonzero expression raised to the zero power is one.
Example 6.63
Simplify:
- \((5 b)^{0}\)
- \(\left(-4 a^{2} b\right)^{0}\).
Solution
|
|
\((5 b)^{0}\) |
| Use the definition of the zero exponent. | \(1\) |
|
|
\(\left(-4 a^{2} b\right)^{0}\) |
| Use the definition of the zero exponent. | \(1\) |
Try It 6.125
Simplify:
- \((11 z)^{0}\)
- \((11pq)^{3}\).
Try It 6.126
Simplify:
- \((-6 d)^{0}\)
- \(\left(-8 m^{2} n^{3}\right)^{0}\).
Simplify Expressions Using the Quotient to a Power Property
Now we will look at an example that will lead us to the Quotient to a Power Property.
| \(\left(\dfrac{x}{y}\right)^{3}\) | |
| This means: | \(\dfrac{x}{y} \cdot \dfrac{x}{y} \cdot \dfrac{x}{y}\) |
| Multiply the fractions. | \(\dfrac{x \cdot x \cdot x}{y \cdot y \cdot y}\) |
| Write with exponents. | \(\dfrac{x^{3}}{y^{3}}\) |
Notice that the exponent applies to both the numerator and the denominator.
| We write: | \(\left(\dfrac{x}{y}\right)^{3}\) |
| \(\dfrac{x^{3}}{y^{3}}\) |
This leads to the Quotient to a Power Property for Exponents.
Quotient to a Power Property for Exponents
If \(a\) and \(b\) are real numbers, \(b \neq 0\), and \(m\) is a counting number, then
\(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}\)
To raise a fraction to a power, raise the numerator and denominator to that power.
An example with numbers may help you understand this property:
\(\begin{aligned}\left(\dfrac{2}{3}\right)^{3} &=\dfrac{2^{3}}{3^{3}} \\ \dfrac{2}{3} \cdot \dfrac{2}{3} \cdot \dfrac{2}{3} &=\dfrac{8}{27} \\ \dfrac{8}{27} &=\dfrac{8}{27}\text{✓} \end{aligned}\)
Example 6.64
Simplify:
- \(\left(\dfrac{3}{7}\right)^{2}\)
- \(\left(\dfrac{b}{3}\right)^{4}\)
- \(\left(\dfrac{k}{j}\right)^{3}\).
Solution
-
\(\left(\dfrac{3}{7}\right)^{2}\) Use the Quotient Property, \(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}\). \(\dfrac{3^{2}}{7^{2}}\) Simplify. \(\dfrac{9}{49}\) -
\(\left(\dfrac{b}{3}\right)^{4}\) \(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}\). \(\dfrac{b^{4}}{3^{4}}\) Simplify. \(\dfrac{b^{4}}{81}\) -
\(\left(\dfrac{k}{j}\right)^{3}\) Raise the numerator and denominator to the third power. \(\dfrac{k^{3}}{j^{3}}\)
Try It 6.127
Simplify:
- \(\left(\dfrac{5}{8}\right)^{2}\)
- \(\left(\dfrac{p}{10}\right)^{4}\)
- \(\left(\dfrac{m}{n}\right)^{7}\).
Try It 6.128
Simplify:
- \(\left(\dfrac{1}{3}\right)^{3}\)
- \(\left(\dfrac{-2}{q}\right)^{3}\)
- \(\left(\dfrac{w}{x}\right)^{4}\).
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 \(a\) 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: \(\dfrac{\left(2 x^{4}\right)^{5}}{\left(4 x^{3}\right)^{2}\left(x^{3}\right)^{5}}\).
Divide Monomials
You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you'll see how to use these properties to divide monomials. Later, you'll use them to divide polynomials.
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: \(\dfrac{27 p^{4} q^{7}}{-45 p^{12} q}\).
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 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: \(\dfrac{30 m^{5} n^{11}}{48 m^{10} n^{14}}\).
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. This follows the order of operations. Remember, a fraction bar is a grouping symbol.
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: \(\dfrac{\left(-12 x^{6} y^{9}\right)\left(-4 x^{5} y^{8}\right)}{-12 x^{10} y^{12}}\).