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# Exponent of a Quotient

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Exponent of a Quotient

What if you had a fractional expression containing exponents that was raised to a secondary power, like $\left ( \frac{x^8}{x^4} \right )^5$ ? How could you simplify it? After completing this Concept, you'll be able to use the power of a quotient property to simplify exponential expressions like this one.

### Guidance

When we raise a whole quotient to a power, another special rule applies. 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 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: $\left ( \frac{x^n}{y^m} \right )^p = \frac{x^{n \cdot p}}{y^{m \cdot p}}$

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

#### Example A

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

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: $\frac{4^5}{4^2} = 4^{5-2} = 4^3 = 64$

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

b) Use the quotient rule first and then evaluate the result: $\frac{5^3}{5^7} = \frac{1}{5^4} = \frac{1}{625}$

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

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: $\left ( \frac{3^4}{5^2} \right )^2 = \frac{3^8}{5^4} = \frac{6561}{625}$

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

#### Example B

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 and then the quotient rule: $\left ( \frac{x^4}{x} \right )^5 = \frac{x^{20}}{x^5} = x^{15}$

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

#### Example C

Simplify the following expressions.

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

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

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

Then reduce the numbers and apply the quotient rule on each fraction to get $3xy$ .

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

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

Watch this video for help with the Examples above.

### Vocabulary

• Quotient of Powers Property: For all real numbers $x$ ,

$\frac{x^n}{x^m} =x^{n-m}$ .

• Power of a Quotient Property:

$\left(\frac{x^n}{y^m}\right)^p = \frac{x^{n \cdot p}}{y^{m \cdot p}}$

### Guided Practice

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

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:

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

Then apply the quotient rule to simplify the fraction:

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

And finally simplify with the product rule:

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

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

Simplify inside the parentheses by reducing the numbers:

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

Then apply the power rule to the first fraction:

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

Group like terms together:

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

And apply the quotient rule to each fraction:

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

### Practice

Evaluate the following expressions.

1. $\left ( \frac{3}{8} \right )^2$
2. $\left ( \frac{2^2}{3^3} \right )^3$
3. $\left ( \frac{2^3 \cdot 4^2}{2^4} \right )^2$

Simplify the following expressions.

1. $\left ( \frac{a^3b^4}{a^2b} \right )^3$
2. $\left ( \frac{18a^4}{15a^{10}} \right )^4$
3. $\left ( \frac{x^6y^2}{x^4y^4} \right )^3$
4. $\left ( \frac{6a^2}{4b^4} \right )^2 \cdot \frac{5b}{3a}$
5. $\frac{(2a^2bc^2)(6abc^3)}{4ab^2c}$
6. $\frac{(2a^2bc^2)(6abc^3)}{4ab^2c}$ for $a=2, b=1,$ and $c=3$
7. $\left ( \frac{3x^2y}{2z} \right )^3 \cdot \frac{z^2}{x}$ for $x=1, y=2,$ and $z=-1$
8. $\frac{2x^3}{xy^2} \cdot \left ( \frac{x}{2y} \right )^2$ for $x=2, y=-3$
9. $\frac{2x^3}{xy^2} \cdot \left ( \frac{x}{2y} \right )^2$ for $x=0, y=6$
10. If $a=2$ and $b=3$ , simplify $\frac{(a^2b)(bc)^3}{a^3c^2}$ as much as possible.