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8.2: Exponent Properties Involving Quotients

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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

You saw in the last section that we can use exponent rules to simplify products of numbers and variables. In this section, you will learn that there are similar rules you can use to simplify quotients. Let’s take an example of a quotient, x^7 divided by x^4.

\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

You should see that when we divide two powers of x, the number of factors of x in the solution is the difference between the factors in the numerator of the fraction, and the factors in the denominator. In other words, when dividing expressions with the same base, keep the base and subtract the exponent in the denominator from the exponent in the numerator.

Quotient Rule for Exponents:  \frac{x^n}{x^m}=x^{n-m}

When we have problems with different bases, we apply the quotient rule separately for each base.

\frac{x^5 y^3} {x^3 y^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^2 y && \text{OR } && \frac{x^5y^3} {x^3y^2} = x^{5 - 3} \cdot y^{3 - 2} = x^2 y

Example 1

Simplify each of the following expressions using the quotient rule.

(a)  \frac{x^{10}} {x^5}

(b)  \frac{a^6} {a}

(c)  \frac{a^5 b^4} {a^3 b^2}

Solution

Apply the quotient rule.

(a)  \frac{x^{10}} {x^5} = x^{10 - 5} = x^5

(b)  \frac{a^6} {a} = a^{6 - 1} = a^5

(c)  \frac{a^5 b^4} {a^3 b^2} = a^{5 - 3} \cdot b^{4 - 2} = a^2 b^2

Now let’s see what happens if the exponent on the denominator is bigger than the exponent in the numerator.

Example 2

& \text{Divide.} \ x^4 \div x^7 && \text{Apply the quotient rule.}\\& \frac{x^4} {x^7} = x^{4 - 7} = x^{-3}

A negative exponent!? What does that mean?

Let’s do the division longhand by writing each term in factored form.

 \frac{x^4} {x^6} = \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 } = \frac{1} {x \cdot x} = \frac{1} {x^2}

We see that when the exponent in the denominator is bigger than the exponent in the numerator, we still subtract the powers. This time we subtract the smaller power from the bigger power and we leave the x's in the denominator.

When you simplify quotients, to get answers with positive exponents you subtract the smaller exponent from the bigger exponent and leave the variable where the bigger power was.

  • We also discovered what a negative power means x^{-3}=\frac{1}{x^3}. We'll learn more on this in the next section!

Example 3

Simplify the following expressions, leaving all powers positive.

(a)  \frac{x^2} {x^6}

(b)  \frac{a^2 b^6} {a^5 b}

Solution

(a) Subtract the exponent in the numerator from the exponent in the denominator and leave the x's in the denominator.

 \frac{x^2} {x^6} = \frac{1} {x^{6 - 2}} = \frac{1} {x^4}

(b) Apply the rule on each variable separately.

 \frac{a^2 b^6} {a^5 b} = \frac{1} {a^{5 - 2}} \cdot \frac{b^{6 - 1}} {1} = \frac{b^5} {a^3}

The Power of a Quotient Property

When we apply a power to a quotient, we can learn another special rule. Here is an example.

 \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}

Notice that the power on the outside of the parenthesis multiplies with the power of the x in the numerator and the power of the y in the denominator. This is called the power of a quotient rule.

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

Simplifying Expressions Involving Quotient Properties of Exponents

Let’s apply the rules we just learned to a few examples.

  • When we have numbers with exponents and not variables with exponents, we evaluate.

Example 4

Simplify the following expressions.

(a)  \frac{4^5} {4^2}

(b)  \frac{5^3} {5^7}

(c)  \left (\frac{3^4} {5^2}\right )^2

Solution

In each of the examples, we want to evaluate the numbers.

(a) Use the quotient rule first.

 \frac{4^5} {4^2} = 4^{5 - 2} = 4^3

Then evaluate the result.

 4^3 = 64

OR

We can evaluate each part separately and then divide.

 \frac{1024} {16} = 64

(b) Use the quotient rule first.

 \frac{5^3} {5^7} = \frac{1} {5^{7 - 3}} = \frac{1} {5^4}

Then evaluate the result.

 \frac{1} {5^4} = \frac{1} {625}

OR

We can evaluate each part separately and then reduce.

 \frac{5^3} {5^7} = \frac{125} {78125} = \frac{1} {625}

It makes more sense to apply the quotient rule first for examples (a) and (b). In this way the numbers we are evaluating are smaller because they are simplified first before applying the power.

(c) Use the power rule for quotients first.

 \left (\frac{3^4} {5^2}\right )^2 = \frac{3^8} {5^4}

Then evaluate the result.

 \frac{3^8} {5^4} = \frac{6561} {625}

OR

We evaluate inside the parenthesis first.

 \left (\frac{3^4} {5^2}\right )^2 = \left (\frac{81} {25}\right )^2

Then apply the power outside the parenthesis.

 \left (\frac{81} {25}\right )^2 = \frac{6561} {625}

When we have just one variable in the expression, then we apply the rules straightforwardly.

Example 5: Simplify the following expressions:

(a)  \frac{x^{12}} {x^5}

(b)  \left (\frac{x^4} {x}\right )^5

Solution:

(a) Use the quotient rule.

 \frac{x^{12}} {x^5} = x^{12 - 5} = x^7

(b) Use the power rule for quotients first.

 \left (\frac{x^4} {x}\right )^5 = \frac{x^{20}} {x^5}

Then apply the quotient rule

 \frac{x^{20}} {x^5} = x^{15}

OR

Use the quotient rule inside the parenthesis first.

 \left (\frac{x^4} {x}\right )^5 = (x^3)^5

Then apply the power rule.

 (x^3)^5 = x^{15}

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

Example 6

Simplify the following expressions.

(a)  \frac{6x^2 y^3} {2xy^2}

(b)  \left (\frac{2a^3b^3} {8a^7b}\right )^2

Solution

(a) We group like terms together.

 \frac{6x^2y^3} {2xy^2} = \frac{6} {2} \cdot \frac{x^2} {x} \cdot \frac{y^3} {y^2}

We reduce the numbers and apply the quotient rule on each grouping.

3xy>

(b) We apply the quotient rule inside the parenthesis first.

 \left (\frac{2a^3b^3} {8a^7b}\right )^2 = \left (\frac{b^2} {4a^4}\right )^2

Apply the power rule for quotients.

 \left (\frac{b^2} {4a^4}\right )^2 = \frac{b^4} {16a^8}

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

Example 7

Simplify the following expressions.

(a)  (x^2)^2 \cdot \frac{x^6} {x^4}

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

Solution

(a) We apply the power rule first on the first parenthesis.

 (x^2)^2 \cdot \frac{x^6} {x^4} = x^4 \cdot \frac{x^6} {x^4}

Then apply the quotient rule to simplify the fraction.

 x^4 \cdot \frac{x^6} {x^4} = x^4 \cdot x^4

Apply the product rule to simplify.

 x^4 \cdot x^2 = x^6

(b) Simplify inside the first parenthesis by reducing the numbers.

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

Then we can apply the power rule on the first parenthesis.

 \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}}

Group like terms together.

 \frac{64a^6} {b^{15}} \cdot \frac{b^2} {a^{16}} = 64 \cdot \frac{a^6} {a^{16}} \cdot \frac{b^2} {b^{15}}

Apply the quotient rule on each fraction.

 64 \cdot \frac{a^6} {a^{16}} \cdot \frac{b^2} {b^{15}} = \frac{64} {a^{10} b^{13}}

Review Questions

Evaluate the following expressions.

  1.  \frac{5^6} {5^2}
  2.  \frac{6^7} {6^3}
  3.  \frac{3^4} {3^{10}}
  4.  \left (\frac{2^2} {3^3}\right )^3

Simplify the following expressions.

  1.  \frac{a^3} {a^2}
  2.  \frac{x^5} {x^9}
  3.  \left (\frac{a^3 b^4} {a^2 b}\right )^3
  4.  \frac{x^6 y^2} {x^2 y^5}
  5.  \frac{6a^3} {2a^2}
  6.  \frac{15x^5} {5x}
  7.  \left (\frac{18a^4} {15a^{10}}\right )^4
  8.  \frac{25yx^6} {20y^5 x^2}
  9.  \left (\frac{x^6 y^2} {x^4 y^4}\right )^3
  10.  \left (\frac{6a^2} {4b^4}\right )^2 \cdot \frac{5b} {3a}
  11.  \frac{(3ab)^2 (4a^3 b^4)^3} {(6a^2 b)^4}
  12.  \frac{(2a^2 bc^2) (6abc^3)} {4ab^2c}

Review Answers

  1. 5^4
  2. 6^4=1296
  3. \frac{1}{3^6}=\frac{1}{729}
  4. \frac{2^6}{3^9}=\frac{64}{19683}
  5. a
  6. \frac{1}{x^4}
  7. a^3 b^9
  8. \frac{x^4}{y^3}
  9.  3a
  10. 3x^4
  11. \frac{1296}{625a^4}
  12.  \frac{5x^4}{4y^4}
  13. \frac{x^6}{y^6}
  14.  \frac{15a^3}{4b^7}
  15. \frac{4a^3 b^{10}}{9}
  16. 3a^2 c^4

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