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Dividing Monomials

When dividing monomials, first divide the coefficients, then divide like variables using the exponent quotient rule.

Let's do an example to practice!

Practice Problem

Solution

$$\begin{align} &\dfrac{64x^9y^{1}1z^3}{8y^{15}z^2}\\ \\ &\bigg(\dfrac{64}{8}\bigg)\bigg(\dfrac{x^9y^{11}z^3}{8y^{15}z^2}\bigg)\\ \\ &8\bigg(\dfrac{x^9y^{11}z^3}{8y^{15}z^2}\bigg)\\ \\ &8x^9y^{(11-15)}z^{(3-2)}\\ \\ &8x^9y^{-4}z\\ \end{align}$$

Dividing Monomials

When you multiply monomials, you multiply the coefficients and then add the exponents of the variables.

To divide, you subtract the exponents. Let's look at an example. $$(5x^3y^{12})\div{(10x^2y^{14})}$$ We divide the coefficients and subtract the exponents in the second monomial from the exponents in the first monomial.

First, we simplify the coefficients. $$\begin{align} (5x^3y^{12})\div{(10x^2y^{14})}&=\frac{5x^3y^{12}}{(10x^2y^{14})}\\ &=\frac{{\color{#8925AE}5}x^3y^{12}}{({\color{#8925AE}{10}}x^2y^{14})}\\ &=\frac{{\color{#8925AE}1}x^3y^{12}}{({\color{#8925AE}{2}}x^2y^{14})} \end{align}$$

Now, let's subtract the exponents for $x$. $$\begin{align} &=\frac{1x^{\color{Blue}3}y^{12}}{(2x^{\color{Blue}2}y^{14})}\\ &=\frac{1x^{\color{Blue}3-2}y^{12}}{(2y^{14})}\\ &=\frac{1x^{\color{Blue}1}y^{12}}{(2y^{14})} \end{align}$$

Finally, we subtract the exponents for $y$ and simplify. $$\begin{align} &=\frac{1x^1y^{\color{green}{12}}}{(2y^{\color{green}{14}})}\\ &=\frac{1x^1y^{\color{green}{12-14}}}{(2)}\\ &=\frac{1}{2}x^1y^{\color{green}{-2}}\\ &=\boxed{\frac{1}{2}xy^{-2}} \end{align}$$

Now, we know how to divide and multiply. An important idea in multiplying and dividing is factors. Factors are numbers or variables that multiply together to get an answer. An example is $2\times3=6$ where $2$ and $3$ are factors.

The factors for a monomial $x^5y^3$ are every combination $x^ay^b$ where $a\le{5}$ and $b\le{3}$. You can see this because if $a=1$ and $b=1$, we could see $$(x^1y^1)\times{x^4y^2}=x^5y^3$$

Now, let's try one.

The factors will be for every combination of $a$ and $b$ where $m\le1$ and $n\le3$.

If we start from the beginning, we have the factor $x^0y^0$ which will also yield the factor $x^1y^3$. $$x^0y^0(x^1y^3)=x^1y^3$$

Next, we can use the factor $x^1y^0$ which will also yield $x^1y^3$. $$x^1y^0(x^0y^3)=x^1y^3$$

This will continue. $$\begin{array}{c|c} m&n&\text{Count}\\ \hline 0&0&1\\ 1&0&2\\ 0&1&3\\ 1&1&4\\ 0&2&5\\ 1&2&6\\ 0&3&7\\ 1&3&8\\ \end{array}$$

There will be $8$pairs.

Now you know how to multiply, divide, and have a basic understanding of factoring monomials. Go ahead and try the practice for this section!

(Video) Multiplication and Division to Simplify Algebraic Expressions

by Sums Made Easy

This video by Sums Made Easy covers monomial multiplication and division.

$\large{\text{Summary}}$

The three steps defined in the video are:

1. Identify the like terms.
2. Group the like terms.
3. Simplify the like terms.

Let's look at an example. $$\frac{10a^2b}{5a^2}$$ First, we identify the like terms. We have constants, $a$'s, and $b$'s.

Next, we group the like terms. $$=\left(\frac{10}{5}\right)\times\left(\frac{a^2}{a^2}\right)\times(b)$$

Finally, we simplify. $$\begin{align} &=2\times a^{2-2}\times b\\ &=2ab \end{align}$$