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# 8.2: Exponent Properties Involving Quotients

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Use the quotient of powers property.
• Use the power of a quotient property.
• Simplify expressions involving quotient properties of exponents.

## Use the Quotient of Powers Property

The rules for simplifying quotients of exponents are a lot like the rules for simplifying products. Let’s look at what happens when we divide \begin{align*}x^7\end{align*} by \begin{align*}x^4\end{align*}:

\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*}

You can see that when we divide two powers of \begin{align*}x\end{align*}, the number of \begin{align*}x\end{align*}’s in the solution is the number of \begin{align*}x\end{align*}’s in the top of the fraction minus the number of \begin{align*}x\end{align*}’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*}

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

\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 \qquad \quad \text{OR} \qquad \quad \frac{x^5y^3}{x^3y^2} = x^{5-3} \cdot y^{3-2} = x^2y\end{align*}

Example 1

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*}

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*}?

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

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*}

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*}’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*} would be equal to \begin{align*}\frac{x^3}{1}\end{align*} (or simply \begin{align*}x^3\end{align*}), \begin{align*}\frac{x^4}{x^7}\end{align*} is equal to \begin{align*}\frac{1}{x^3}\end{align*}. And you can also see that \begin{align*}\frac{1}{x^3}\end{align*} is equal to \begin{align*}x^{-3}\end{align*}. We’ll learn more about negative exponents shortly.

Example 2

Simplify the following expressions, leaving all exponents positive.

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

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*}

## The Power of a Quotient Property

When we raise a whole quotient to a power, another special rule applies. Here is an example:

\begin{align*}\left ( \frac{x^3}{y^2} \right )^4 = \left ( \frac{x^3}{y^2} \right ) \cdot \left ( \frac{x^3}{y^2} \right ) \cdot \left ( \frac{x^3}{y^2} \right ) \cdot \left ( \frac{x^3}{y^2} \right ) = \frac{(x \cdot x \cdot x) \cdot (x \cdot x \cdot x) \cdot (x \cdot x \cdot x) \cdot (x \cdot x \cdot x)}{(y \cdot y) \cdot (y \cdot y) \cdot (y \cdot y) \cdot (y \cdot y)} = \frac{x^{12}}{y^8}\end{align*}

Notice that the exponent outside the parentheses is multiplied by the exponent in the numerator and the exponent in the denominator, separately. This is called the power of a quotient rule:

Power Rule for Quotients: \begin{align*}\left ( \frac{x^n}{y^m} \right )^p = \frac{x^{n \cdot p}}{y^{m \cdot p}}\end{align*}

Let’s apply these new rules to a few examples.

Example 3

Simplify the following expressions.

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

b) \begin{align*}\frac{5^3}{5^7}\end{align*}

c) \begin{align*}\left ( \frac{3^4}{5^2} \right )^2\end{align*}

Solution

Since there are just numbers and no variables, we can evaluate the expressions and get rid of the exponents completely.

a) We can use the quotient rule first and then evaluate the result: \begin{align*}\frac{4^5}{4^2} = 4^{5-2} = 4^3 = 64\end{align*}

OR we can evaluate each part separately and then divide: \begin{align*}\frac{4^5}{4^2} = \frac{1024}{16} = 64\end{align*}

b) Use the quotient rule first and hen evaluate the result: \begin{align*}\frac{5^3}{5^7} = \frac{1}{5^4} = \frac{1}{625}\end{align*}

OR evaluate each part separately and then reduce: \begin{align*}\frac{5^3}{5^7} = \frac{125}{78125} = \frac{1}{625}\end{align*}

Notice that it makes more sense to apply the quotient rule first for examples (a) and (b). Applying the exponent rules to simplify the expression before plugging in actual numbers means that we end up with smaller, easier numbers to work with.

c) Use the power rule for quotients first and then evaluate the result: \begin{align*}\left ( \frac{3^4}{5^2} \right )^2 = \frac{3^8}{5^4} = \frac{6561}{625}\end{align*}

OR evaluate inside the parentheses first and then apply the exponent: \begin{align*}\left ( \frac{3^4}{5^2} \right )^2 = \left ( \frac{81}{25} \right )^2 = \frac{6561}{625}\end{align*}

Example 4

Simplify the following expressions:

a) \begin{align*}\frac{x^{12}}{x^5}\end{align*}

b) \begin{align*}\left ( \frac{x^4}{x} \right )^5\end{align*}

Solution

a) Use the quotient rule: \begin{align*}\frac{x^{12}}{x^5} = x^{12-5} = x^7\end{align*}

b) Use the power rule for quotients and then the quotient rule: \begin{align*}\left ( \frac{x^4}{x} \right )^5 = \frac{x^{20}}{x^5} = x^{15}\end{align*}

OR use the quotient rule inside the parentheses first, then apply the power rule: \begin{align*}\left ( \frac{x^4}{x} \right )^5 = (x^3)^5 = x^{15}\end{align*}

Example 5

Simplify the following expressions.

a) \begin{align*}\frac{6x^2y^3}{2xy^2}\end{align*}

b) \begin{align*}\left ( \frac{2a^3b^3}{8a^7b} \right )^2\end{align*}

Solution

When we have a mix of numbers and variables, we apply the rules to each number or each variable separately.

a) Group like terms together: \begin{align*}\frac{6x^2y^3}{2xy^2} = \frac{6}{2} \cdot \frac{x^2}{x} \cdot \frac{y^3}{y^2}\end{align*}

Then reduce the numbers and apply the quotient rule on each fraction to get \begin{align*}3xy\end{align*}.

b) Apply the quotient rule inside the parentheses first: \begin{align*}\left ( \frac{2a^3b^3}{8a^7b} \right )^2 = \left ( \frac{b^2}{4a^4} \right )^2\end{align*}

Then apply the power rule for quotients: \begin{align*}\left ( \frac{b^2}{4a^4} \right )^2 = \frac{b^4}{16a^8}\end{align*}

Example 6

Simplify the following expressions.

a) \begin{align*}(x^2)^2 \cdot \frac{x^6}{x^4}\end{align*}

b) \begin{align*}\left ( \frac{16a^2}{4b^5} \right )^3 \cdot \frac{b^2}{a^{16}}\end{align*}

Solution

In problems where we need to apply several rules together, we must keep the order of operations in mind.

a) We apply the power rule first on the first term:

\begin{align*}(x^2)^2 \cdot \frac{x^6}{x^4} = x^4 \cdot \frac{x^6}{x^4}\end{align*}

Then apply the quotient rule to simplify the fraction:

\begin{align*}x^4 \cdot \frac{x^6}{x^4} = x^4 \cdot x^2\end{align*}

And finally simplify with the product rule:

\begin{align*}x^4 \cdot x^2 = x^6\end{align*}

b) \begin{align*}\left ( \frac{16a^2}{4b^5} \right )^3 \cdot \frac{b^2}{a^{16}}\end{align*}

Simplify inside the parentheses by reducing the numbers:

\begin{align*}\left ( \frac{4a^2}{b^5} \right )^3 \cdot \frac{b^2}{a^{16}}\end{align*}

Then apply the power rule to the first fraction:

\begin{align*}\left ( \frac{4a^2}{b^5} \right )^3 \cdot \frac{b^2}{a^{16}} = \frac{64a^6}{b^{15}} \cdot \frac{b^2}{a^{16}}\end{align*}

Group like terms together:

\begin{align*}\frac{64a^6}{b^{15}} \cdot \frac{b^2}{a^{16}} = 64 \cdot \frac{a^6}{a^{16}} \cdot \frac{b^2}{b^{15}}\end{align*}

And apply the quotient rule to each fraction:

\begin{align*}64 \cdot \frac{a^6}{a^{16}} \cdot \frac{b^2}{b^{15}} = \frac{64}{a^{10}b^{13}}\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*}\left ( \frac{3}{8} \right )^2\end{align*}
5. \begin{align*}\left ( \frac{2^2}{3^3} \right )^3\end{align*}
6. \begin{align*}\frac{2^2 \cdot 3^2}{5^2}\end{align*}
7. \begin{align*}\frac{3^3 \cdot 5^2}{3^7}\end{align*}
8. \begin{align*}\left ( \frac{2^3 \cdot 4^2}{2^4} \right )^2\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*}\left ( \frac{a^3b^4}{a^2b} \right )^3\end{align*}
4. \begin{align*}\frac{x^6y^2}{x^2y^5}\end{align*}
5. \begin{align*}\frac{6a^3}{2a^2}\end{align*}
6. \begin{align*}\frac{15x^5}{5x}\end{align*}
7. \begin{align*}\left ( \frac{18a^4}{15a^{10}} \right )^4\end{align*}
8. \begin{align*}\frac{25yx^6}{20y^5x^2}\end{align*}
9. \begin{align*}\left ( \frac{x^6y^2}{x^4y^4} \right )^3\end{align*}
10. \begin{align*}\left ( \frac{6a^2}{4b^4} \right )^2 \cdot \frac{5b}{3a}\end{align*}
11. \begin{align*}\frac{(3ab)^2(4a^3b^4)^3}{(6a^2b)^4}\end{align*}
12. \begin{align*}\frac{(2a^2bc^2)(6abc^3)}{4ab^2c}\end{align*}
13. \begin{align*}\frac{(2a^2bc^2)(6abc^3)}{4ab^2c}\end{align*} for \begin{align*}a=2, b=1,\end{align*} and \begin{align*}c=3\end{align*}
14. \begin{align*}\left ( \frac{3x^2y}{2z} \right )^3 \cdot \frac{z^2}{x}\end{align*} for \begin{align*}x=1, y=2,\end{align*} and \begin{align*}z=-1\end{align*}
15. \begin{align*}\frac{2x^3}{xy^2} \cdot \left ( \frac{x}{2y} \right )^2\end{align*} for \begin{align*}x=2, y=-3\end{align*}
16. \begin{align*}\frac{2x^3}{xy^2} \cdot \left ( \frac{x}{2y} \right )^2\end{align*} for \begin{align*}x=0, y=6\end{align*}
17. If \begin{align*}a=2\end{align*} and \begin{align*}b=3\end{align*}, simplify \begin{align*}\frac{(a^2b)(bc)^3}{a^3c^2}\end{align*} as much as possible.

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