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8.3: Exponential Properties Involving Quotients

Difficulty Level: At Grade Created by: CK-12
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What if you had a fractional expression like \begin{align*}\frac{x^5}{x^2}\end{align*}x5x2 in which both the numerator and denominator contained exponents? How could you simplify it? After completing this Concept, you'll be able to use the quotient of powers property to simplify exponential expressions like this one.

Watch This

CK-12 Foundation: 0803S Quotient of Powers

Guidance

The rules for simplifying quotients of exponents are a lot like the rules for simplifying products.

Example A

Let’s look at what happens when we divide \begin{align*}x^7\end{align*}x7 by \begin{align*}x^4\end{align*}x4:

\begin{align*} \frac{x^7}{x^4} = \frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot x}{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot \cancel{x}} = \frac{x \cdot x \cdot x}{1} = x^3\end{align*}x7x4=xxxxxxxxxxx=xxx1=x3

You can see that when we divide two powers of \begin{align*}x\end{align*}x, the number of \begin{align*}x\end{align*}x’s in the solution is the number of \begin{align*}x\end{align*}x’s in the top of the fraction minus the number of \begin{align*}x\end{align*}x’s in the bottom. In other words, when dividing expressions with the same base, we keep the same base and simply subtract the exponent in the denominator from the exponent in the numerator.

Quotient Rule for Exponents: \begin{align*}\frac{x^n}{x^m} = x^{(n-m)}\end{align*}xnxm=x(nm)

When we have expressions with more than one base, we apply the quotient rule separately for each base:

Now let’s see what happens if the exponent in the denominator is bigger than the exponent in the numerator. For example, what happens when we apply the quotient rule to \begin{align*}\frac{x^4}{x^7}\end{align*}x4x7?

The quotient rule tells us to subtract the exponents. 4 minus 7 is -3, so our answer is \begin{align*}x^{-3}\end{align*}x3. A negative exponent! What does that mean?

Example B

\begin{align*}\frac{x^5y^3}{x^3y^2}=\frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot x \cdot x}{\cancel{x} \cdot \cancel{x} \cdot \cancel{x}} \cdot \frac{\cancel{y} \cdot \cancel{y} \cdot y}{\cancel{y} \cdot \cancel{y}} = \frac{x \cdot x}{1} \cdot \frac{y}{1} = x^2y \end{align*}x5y3x3y2=xxxxxxxxyyyyy=xx1y1=x2y

OR

\begin{align*}\frac{x^5y^3}{x^3y^2} = x^{5-3} \cdot y^{3-2} = x^2y\end{align*}x5y3x3y2=x53y32=x2y

Well, let’s look at what we get when we do the division longhand by writing each term in factored form:

\begin{align*}\frac{x^4}{x^7} = \frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot \cancel{x}}{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot x} = \frac{1}{x \cdot x \cdot x} = \frac{1}{x^3}\end{align*}x4x7=xxxxxxxxxxx=1xxx=1x3

Even when the exponent in the denominator is bigger than the exponent in the numerator, we can still subtract the powers. The \begin{align*}x\end{align*}x’s that are left over after the others have been canceled out just end up in the denominator instead of the numerator. Just as \begin{align*}\frac{x^7}{x^4}\end{align*}x7x4 would be equal to \begin{align*}\frac{x^3}{1}\end{align*}x31 (or simply \begin{align*}x^3\end{align*}x3), \begin{align*}\frac{x^4}{x^7}\end{align*}x4x7 is equal to \begin{align*}\frac{1}{x^3}\end{align*}1x3. And you can also see that \begin{align*}\frac{1}{x^3}\end{align*}1x3 is equal to \begin{align*}x^{-3}\end{align*}x3. We’ll learn more about negative exponents shortly.

Example C

Simplify the following expressions, leaving all exponents positive.

a) \begin{align*}\frac{x^2}{x^6}\end{align*}x2x6

b) \begin{align*}\frac{a^2b^6}{a^5b}\end{align*}

Solution

a) Subtract the exponent in the numerator from the exponent in the denominator and leave the \begin{align*}x\end{align*}’s in the denominator: \begin{align*}\frac{x^2}{x^6} = \frac{1}{x^{6-2}}= \frac{1}{x^4}\end{align*}

b) Apply the rule to each variable separately: \begin{align*}\frac{a^2b^6}{a^5b} = \frac{1}{a^{5-2}} \cdot \frac{b^{6-1}}{1} = \frac{b^5}{a^3}\end{align*}

Watch this video for help with the Examples above.

CK-12 Foundation: Quotient of Powers

Vocabulary

Quotient of Powers Property: For all real numbers \begin{align*}x\end{align*},

\begin{align*}\frac{x^n}{x^m} =x^{n-m}\end{align*}.

Guided Practice

Simplify each of the following expressions using the quotient rule.

a) \begin{align*}\frac{x^{10}}{x^5}\end{align*}

b) \begin{align*}\frac{a^6}{a}\end{align*}

c) \begin{align*}\frac{a^5b^4}{a^3b^2}\end{align*}

Solution

a) \begin{align*}\frac{x^{10}}{x^5}= x^{10-5} = x^5\end{align*}

b) \begin{align*}\frac{a^6}{a} = a^{6-1} =a^5\end{align*}

c) \begin{align*}\frac{a^5b^4}{a^3b^2}= a^{5-3} \cdot b^{4-2} = a^2b^2\end{align*}

Review Questions

Evaluate the following expressions.

  1. \begin{align*}\frac{5^6}{5^2}\end{align*}
  2. \begin{align*}\frac{6^7}{6^3}\end{align*}
  3. \begin{align*}\frac{3^4}{3^{10}}\end{align*}
  4. \begin{align*}\frac{2^2 \cdot 3^2}{5^2}\end{align*}
  5. \begin{align*}\frac{3^3 \cdot 5^2}{3^7}\end{align*}

Simplify the following expressions.

  1. \begin{align*}\frac{a^3}{a^2}\end{align*}
  2. \begin{align*}\frac{x^5}{x^9}\end{align*}
  3. \begin{align*}\frac{x^6y^2}{x^2y^5}\end{align*}
  4. \begin{align*}\frac{6a^3}{2a^2}\end{align*}
  5. \begin{align*}\frac{15x^5}{5x}\end{align*}
  6. \begin{align*}\frac{25yx^6}{20y^5x^2}\end{align*}

Vocabulary

Base

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression 32^4, 32 is the base, and 4 is the exponent.
Exponent

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.
Power

Power

The "power" refers to the value of the exponent. For example, 3^4 is "three to the fourth power".

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