Divide Monomials

Simplify expressions using the Quotient Property of Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize them here.

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have already learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us 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 $\tfrac{x^2}{x^3}$. Writing out the factors:

Notice that in each case the bases were the same and we subtracted the exponents.

• When the larger exponent was in the numerator, we were left with factors in the numerator and $1$ in the denominator, which we simplified.
• When the larger exponent was in the denominator, we were left with factors in the denominator, and $1$ in the numerator, which could not be simplified.

So we write $\tfrac{x^5}{x^2} = x^{5-2} = x^3$ and $\tfrac{x^2}{x^3} = \tfrac{1}{x^{3-2}} = \tfrac{1}{x}$.

A couple of examples with numbers may help to verify this property. Since $3^4 / 3^2 \stackrel{?}{=} 3^{4-2}$, checking: $81/9 \stackrel{?}{=} 3^2$, and indeed $9 = 9$. Since $5^2 / 5^3 \stackrel{?}{=} 1/5^{3-2}$, checking: $25/125 \stackrel{?}{=} 1/5$, and indeed $\tfrac{1}{5} = \tfrac{1}{5}$. When we work with numbers and the exponent is less than or equal to $3$, we will apply the exponent. When the exponent is greater than $3$, we leave the answer in exponential form.

Example. Simplify: (a) $\tfrac{x^{10}}{x^8}$ (b) $\tfrac{2^9}{2^2}$.

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

(a) Since $10 > 8$, there are more factors of $x$ in the numerator, so we use the quotient property with $m > n$: $\tfrac{x^{10}}{x^8} = x^{10-8} = x^2$.

(b) Since $9 > 2$, there are more factors of $2$ in the numerator, so we use the quotient property with $m > n$: $\tfrac{2^9}{2^2} = 2^{9-2} = 2^7$. Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

Example. Simplify: (a) $\tfrac{b^{10}}{b^{15}}$ (b) $\tfrac{3^3}{3^5}$.

(a) Since $15 > 10$, there are more factors of $b$ in the denominator, so we use the quotient property with $n > m$: $\tfrac{b^{10}}{b^{15}} = \tfrac{1}{b^{15-10}} = \tfrac{1}{b^5}$.

(b) Since $5 > 3$, there are more factors of $3$ in the denominator: $\tfrac{3^3}{3^5} = \tfrac{1}{3^{5-3}} = \tfrac{1}{3^2} = \tfrac{1}{9}$. Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and $1$ in the numerator.

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

(a) Since $9 > 5$, there are more $a$’s in the denominator, so we end up with factors in the denominator: $\tfrac{a^5}{a^9} = \tfrac{1}{a^{9-5}} = \tfrac{1}{a^4}$.

(b) Since $11 > 7$, there are more factors of $x$ in the numerator: $\tfrac{x^{11}}{x^7} = x^{11-7} = x^4$.

Simplify expressions with zero exponents

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 earlier work with fractions, we 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 of Exponents shows us how to simplify $\tfrac{a^m}{a^n}$ when $m > n$ and when $n < m$. What if $m = n$?

Now we will simplify $\tfrac{a^m}{a^m}$ in two ways to lead us to the definition of the zero exponent. Consider first $\tfrac{8}{8}$, which we know is $1$. Writing $8$ as $2^3$: $\tfrac{2^3}{2^3} = 1$. Subtracting exponents: $2^{3-3} = 1$. Simplifying: $2^0 = 1$.

In general, for $a \neq 0$, the pattern $\tfrac{a^m}{a^m} = a^{m-m} = a^0$ counts $m$ matching factors of $a$ in the numerator against $m$ matching factors of $a$ in the denominator, which cancel completely to $1$. We see $\tfrac{a^m}{a^m}$ simplifies to both $a^0$ and to $1$. So $a^0 = 1$.

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

Example. Simplify: (a) $12^0$ (b) $y^0$.

The definition says any non-zero number raised to the zero power is $1$. So $12^0 = 1$ and $y^0 = 1$.

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 Rule to rewrite this expression: using the rule, $(2x)^0 = 2^0 x^0$; using the Zero Exponent Property, $2^0 x^0 = 1 \cdot 1$; simplifying, this is $1$. This tells us that any non-zero expression raised to the zero power is one.

Example. Simplify: $(7z)^0$.

Using the definition of the zero exponent, $(7z)^0 = 1$.

Example. Simplify: (a) $(-3x^2y)^0$ (b) $-3x^2y^0$.

(a) Here the entire product is raised to the zero power, so $(-3x^2y)^0 = 1$.

(b) Here, notice that only the variable $y$ is being raised to the zero power — the exponent applies only to $y$, not to the whole expression. Using the definition of the zero exponent on just that factor: $-3x^2 y^0 = -3x^2 \cdot 1 = -3x^2$.

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 $\left(\tfrac{x}{y}\right)^3$. This means $\tfrac{x}{y} \cdot \tfrac{x}{y} \cdot \tfrac{x}{y}$. Multiplying the fractions gives $\tfrac{x \cdot x \cdot x}{y \cdot y \cdot y}$, which written with exponents is $\tfrac{x^3}{y^3}$.

Notice that the exponent applies to both the numerator and the denominator. We see that $\left(\tfrac{x}{y}\right)^3$ is $\tfrac{x^3}{y^3}$. This leads to the Quotient to a Power Property for Exponents.

An example with numbers may help you understand this property: $\left(\tfrac{2}{3}\right)^3 \stackrel{?}{=} \tfrac{2^3}{3^3}$. Checking directly, $\tfrac{2}{3} \cdot \tfrac{2}{3} \cdot \tfrac{2}{3} \stackrel{?}{=} \tfrac{8}{27}$, and indeed $\tfrac{8}{27} = \tfrac{8}{27}$.

Example. Simplify: (a) $\left(\tfrac{5}{8}\right)^2$ (b) $\left(\tfrac{x}{3}\right)^4$ (c) $\left(\tfrac{y}{m}\right)^3$.

(a) Using the Quotient to a Power Property, $\left(\tfrac{5}{8}\right)^2 = \tfrac{5^2}{8^2} = \tfrac{25}{64}$.

(b) Using the Quotient to a Power Property, $\left(\tfrac{x}{3}\right)^4 = \tfrac{x^4}{3^4} = \tfrac{x^4}{81}$.

(c) Raising the numerator and denominator to the third power, $\left(\tfrac{y}{m}\right)^3 = \tfrac{y^3}{m^3}$.

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: $\tfrac{(x^2)^3}{x^5}$.

Multiplying the exponents in the numerator using the Power Property gives $\tfrac{x^6}{x^5}$. Subtracting the exponents gives $x$.

Example. Simplify: $\tfrac{m^8}{(m^2)^4}$.

Multiplying the exponents in the denominator using the Power Property gives $\tfrac{m^8}{m^8}$. Subtracting the exponents gives $m^0$. Using the zero power property gives $1$.

Example. Simplify: $\left(\tfrac{x^7}{x^3}\right)^2$.

Remember parentheses come before exponents, and the bases are the same so we can simplify inside the parentheses first — subtracting the exponents gives $(x^{7-3})^2$. Simplifying gives $(x^4)^2$. Multiplying the exponents gives $x^8$.

Example. Simplify: $\left(\tfrac{p^2}{q^5}\right)^3$.

Here we cannot simplify inside the parentheses first, since the bases are not the same. Instead we raise the numerator and denominator to the third power using the Quotient to a Power Property, giving $\tfrac{(p^2)^3}{(q^5)^3}$. Using the Power Property gives $\tfrac{p^6}{q^{15}}$.

Example. Simplify: $\left(\tfrac{2x^3}{3y}\right)^4$.

Raising the numerator and denominator to the fourth power using the Quotient to a Power Property gives $\tfrac{(2x^3)^4}{(3y)^4}$. Raising each factor to the fourth power using the Product to a Power Property gives $\tfrac{2^4 (x^3)^4}{3^4 y^4}$. Using the Power Property and simplifying gives $\tfrac{16x^{12}}{81y^4}$.

Example. Simplify: $\tfrac{(y^2)^3 (y^2)^4}{(y^5)^4}$.

Using the Power Property throughout gives $\tfrac{(y^6)(y^8)}{y^{20}}$. Adding the exponents in the numerator using the Product Property gives $\tfrac{y^{14}}{y^{20}}$. Using the Quotient Property gives $\tfrac{1}{y^6}$.

Divide monomials

We have now seen all the properties of exponents. We’ll use them to divide monomials. Later, you’ll use them to divide polynomials.

Example. Find the quotient: $56x^5 \div 7x^2$.

Rewriting as a fraction gives $\tfrac{56x^5}{7x^2}$. Using fraction multiplication to separate the number part from the variable part gives $\tfrac{56}{7} \cdot \tfrac{x^5}{x^2}$. Using the Quotient Property gives $8x^3$.

Example. Find the quotient: $\tfrac{42x^2y^3}{-7xy^5}$.

Using fraction multiplication gives $\tfrac{42}{-7} \cdot \tfrac{x^2}{x} \cdot \tfrac{y^3}{y^5}$. Simplifying and using the Quotient Property gives $-6 \cdot x \cdot \tfrac{1}{y^2}$. Multiplying gives $-\tfrac{6x}{y^2}$.

Example. Find the quotient: $\tfrac{24a^5b^3}{48ab^4}$.

Using fraction multiplication gives $\tfrac{24}{48} \cdot \tfrac{a^5}{a} \cdot \tfrac{b^3}{b^4}$. Simplifying and using the Quotient Property gives $\tfrac{1}{2} \cdot a^4 \cdot \tfrac{1}{b}$. Multiplying gives $\tfrac{a^4}{2b}$.

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. Find the quotient: $\tfrac{14x^7y^{12}}{21x^{11}y^6}$.

Simplifying directly and using the Quotient Property gives $\tfrac{2y^6}{3x^4}$. Be very careful to simplify $\tfrac{14}{21}$ by dividing out a common factor, and to simplify the variables by subtracting their exponents.

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.

Example. Find the quotient: $\tfrac{(3x^3y^2)(10x^2y^3)}{6x^4y^5}$.

Remember, the fraction bar is a grouping symbol. We will simplify the numerator first. Simplifying the numerator gives $\tfrac{30x^5y^5}{6x^4y^5}$. Simplifying, using the Quotient Rule, gives $5x$.

Key terms

Quotient Property of Exponents — for a real number $a \neq 0$ and whole numbers $m, n$: $\tfrac{a^m}{a^n} = a^{m-n}$ when $m > n$, and $\tfrac{a^m}{a^n} = \tfrac{1}{a^{n-m}}$ when $n > m$. zero exponent — any nonzero number or expression raised to the power $0$ equals $1$. Quotient to a Power Property — $\left(\tfrac{a}{b}\right)^m = \tfrac{a^m}{b^m}$ for $b \neq 0$. monomial — a single term made of a coefficient and variables raised to whole-number powers, divided using these exponent properties one variable at a time.

This section is adapted from Prealgebra 2e, Section 10.4: 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: 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.