Divide Monomials

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.

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.

As before, we’ll try to discover a property by looking at some examples. Consider $\tfrac{x^5}{x^2}$ and $\tfrac{x^2}{x^3}$. What do they mean?

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, and $\tfrac{x^5}{x^2} = x^{5-2} = x^3$. When the larger exponent was in the denominator, we were left with factors in the denominator — notice the numerator of $1$ — and $\tfrac{x^2}{x^3} = \tfrac{1}{x^{3-2}} = \tfrac{1}{x}$. This leads to the Quotient Property for Exponents.

A couple of examples with numbers may help to verify this property: $\tfrac{3^4}{3^2} \overset{?}{=} 3^{4-2}$, so $\tfrac{81}{9} \overset{?}{=} 3^2$, giving $9 = 9$ ✓; and $\tfrac{5^2}{5^3} \overset{?}{=} \tfrac{1}{5^{3-2}}$, so $\tfrac{25}{125} \overset{?}{=} \tfrac{1}{5^1}$, giving $\tfrac{1}{5} = \tfrac{1}{5}$ ✓.

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Example. Simplify: (a) $\tfrac{x^9}{x^7}$ (b) $\tfrac{3^{10}}{3^2}$.

(a) Since $9 > 7$, there are more factors of $x$ in the numerator. Use the Quotient Property, $\tfrac{a^m}{a^n} = a^{m-n}$, and simplify: $\tfrac{x^9}{x^7} = x^{9-7} = x^2$.

(b) Since $10 > 2$, there are more factors of $3$ in the numerator. Use the Quotient Property and simplify: $\tfrac{3^{10}}{3^2} = 3^{10-2} = 3^8$.

Example. Simplify: (a) $\tfrac{b^8}{b^{12}}$ (b) $\tfrac{7^3}{7^5}$.

(a) Since $12 > 8$, there are more factors of $b$ in the denominator. Use the Quotient Property, $\tfrac{a^m}{a^n} = \tfrac{1}{a^{n-m}}$, and simplify: $\tfrac{b^8}{b^{12}} = \tfrac{1}{b^{12-8}} = \tfrac{1}{b^4}$.

(b) Since $5 > 3$, there are more factors of $3$ in the denominator. Use the Quotient Property and simplify: $\tfrac{7^3}{7^5} = \tfrac{1}{7^{5-3}} = \tfrac{1}{7^2} = \tfrac{1}{49}$.

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. Simplify: (a) $\tfrac{a^5}{a^9}$ (b) $\tfrac{x^{11}}{x^7}$.

(a) 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: $\tfrac{a^5}{a^9} = \tfrac{1}{a^{9-5}} = \tfrac{1}{a^4}$.

(b) Notice there are more factors of $x$ in the numerator, since $11 > 7$. So we will end up with factors in the numerator: $\tfrac{x^{11}}{x^7} = x^{11-7} = x^4$.

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 $\tfrac{a^m}{a^m}$. From your earlier work with fractions, you know that $\tfrac{2}{2} = 1$, $\tfrac{17}{17} = 1$, and $\tfrac{-43}{-43} = 1$. In words, a number divided by itself is $1$. So $\tfrac{x}{x} = 1$, for any $x$ ($x \neq 0$), since any number divided by itself is $1$.

The Quotient Property for Exponents shows us how to simplify $\tfrac{a^m}{a^n}$ when $m > n$ and when $n < m$ by subtracting exponents. What if $m = n$? Consider $\tfrac{8}{8}$, which we know is $1$:

We wrote $8$ as $2^3$, subtracted exponents, and simplified. Now we will simplify $\tfrac{a^m}{a^m}$ in two ways to lead us to the definition of the zero exponent. On the one hand, $\tfrac{a^m}{a^m} = a^{m-m} = a^0$; on the other hand, $\tfrac{a^m}{a^m} = 1$ since any nonzero quantity divided by itself is $1$. So $a^0 = 1$.

In this text, we assume any variable that we raise to the zero power is not zero.

Example. Simplify: (a) $9^0$ (b) $n^0$.

The definition says any non-zero number raised to the zero power is $1$.

(a) Use the definition of the zero exponent: $9^0 = 1$.

(b) Use the definition of the zero exponent: $n^0 = 1$.

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 $(2x)^0$. We can use the Product to a Power Property to rewrite this expression: $(2x)^0 = 2^0 x^0 = 1 \cdot 1 = 1$. This tells us that any nonzero expression raised to the zero power is one.

Example. Simplify: (a) $(5b)^0$ (b) $\left(-4a^2 b\right)^0$.

(a) Use the definition of the zero exponent: $(5b)^0 = 1$.

(b) Use the definition of the zero exponent: $\left(-4a^2 b\right)^0 = 1$.

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. Consider $\left(\tfrac{x}{y}\right)^3$. What does this mean?

Notice that the exponent applies to both the numerator and the denominator. This leads to the Quotient to a Power Property for Exponents.

An example with numbers may help you understand this property: $\left(\tfrac{2}{3}\right)^3 \overset{?}{=} \tfrac{2^3}{3^3}$, and since $\tfrac{2}{3} \cdot \tfrac{2}{3} \cdot \tfrac{2}{3} = \tfrac{8}{27}$, we have $\tfrac{8}{27} = \tfrac{8}{27}$ ✓.

Example. Simplify: (a) $\left(\tfrac{3}{7}\right)^2$ (b) $\left(\tfrac{b}{3}\right)^4$ (c) $\left(\tfrac{k}{j}\right)^3$.

(a) Use the Quotient to a Power Property, then simplify: $\left(\tfrac{3}{7}\right)^2 = \tfrac{3^2}{7^2} = \tfrac{9}{49}$.

(b) Use the Quotient to a Power Property, then simplify: $\left(\tfrac{b}{3}\right)^4 = \tfrac{b^4}{3^4} = \tfrac{b^4}{81}$.

(c) Raise the numerator and denominator to the third power: $\left(\tfrac{k}{j}\right)^3 = \tfrac{k^3}{j^3}$.

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.

Example. Simplify:

Multiply the exponents in the numerator using the Power Property, then subtract the exponents using the Quotient Property:

Example. Simplify:

Remember parentheses come before exponents. Notice the bases are the same, so we can simplify inside the parentheses first by subtracting the exponents; then multiply the exponents using the Power Property:

Example. Simplify:

Here we cannot simplify inside the parentheses first, since the bases are not the same. Raise the numerator and denominator to the fourth power using the Quotient to a Power Property, then use the Power Property and simplify:

Example. Simplify:

Raise the numerator and denominator to the fourth power using the Quotient to a Power Property, then raise each factor to the fourth power and use the Power Property to simplify:

Example. Simplify:

Use the Power Property on each factor, then add the exponents in the numerator using the Product Property, then use the Quotient Property to simplify:

Example. Simplify:

Use the Product to a Power Property, then the Power Property, then add the exponents in the denominator using the Product Property, and finally use the Quotient Property and simplify:

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. Find the quotient: $56x^7 \div 8x^3$.

Rewrite as a fraction, use fraction multiplication to separate the numbers from the variables, then simplify and use the Quotient Property:

Example. Find the quotient:

Use fraction multiplication to separate the numbers from each variable, then simplify using the Quotient Property and multiply:

Example. Find the quotient:

Use fraction multiplication, then simplify using the Quotient Property and multiply:

Once you become familiar with the process, you may be able to simplify a fraction in one step.

Example. Find the quotient:

Be very careful to simplify $\tfrac{14}{21}$ by dividing out a common factor, and to simplify the variables by subtracting their exponents:

In the 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, since a fraction bar is a grouping symbol.

Example. Find the quotient:

Simplify the numerator by multiplying the two monomials, then simplify the fraction:

Key terms

Quotient Property for Exponents — to divide powers with like bases, subtract the exponents: $\tfrac{a^m}{a^n} = a^{m-n}$ when $m > n$, and $\tfrac{a^m}{a^n} = \tfrac{1}{a^{n-m}}$ when $n > m$ (with $a \neq 0$). Zero Exponent — any nonzero number or expression raised to the zero power is $1$: $a^0 = 1$ for $a \neq 0$. Quotient to a Power Property for Exponents — to raise a fraction to a power, raise the numerator and denominator to that power: $\left(\tfrac{a}{b}\right)^m = \tfrac{a^m}{b^m}$ (with $b \neq 0$). monomial — a one-term algebraic expression, such as $7x^4$, that we can divide by applying these properties of exponents.

This section is adapted from Elementary Algebra 2e, Section 6.5: Divide Monomials by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the worked-example step tables as prose and typeset equations; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.