# Exponential Properties Involving Quotients

## Key Questions

• $\frac{{a}^{m}}{{a}^{n}} = {a}^{m - n}$

This property allows you to simplify problems where you have a fraction of the same numbers ( $a$) raised to different powers ($m \mathmr{and} n$).
For example:

$\frac{{3}^{3}}{{3}^{2}} = \frac{3 \cdot 3 \cdot 3}{3 \cdot 3} = {3}^{3 - 2} = 3$

You can see how the power of 3, in the numerator, is "reduced" by the presence of the power 2 in the denominator.

You can also check te result by doing the multiplications:

$\frac{{3}^{3}}{{3}^{2}} = \frac{3 \cdot 3 \cdot 3}{3 \cdot 3} = \frac{27}{9} = 3$

As a challenge try to find out what happens when $m = n$ !!!!!

• The Power of a Quotient Rule states that the power of a quotient is equal to the quotient obtained when the numerator and denominator are each raised to the indicated power separately, before the division is performed.
i.e.: ${\left(\frac{a}{b}\right)}^{n} = {a}^{n} / {b}^{n}$
For example:
${\left(\frac{3}{2}\right)}^{2} = {3}^{2} / {2}^{2} = \frac{9}{4}$

You can test this rule by using numbers that are easy to manipulate:
Consider: $\frac{4}{2}$ (ok it is equal to $2$ but for the moment let it stay as a fraction), and let us calculate it with our rule first:
${\left(\frac{4}{2}\right)}^{2} = {4}^{2} / {2}^{2} = \frac{16}{4} = 4$
Let us, now, solve the fraction first and then raise to the power of $2$:
${\left(\frac{4}{2}\right)}^{2} = {\left(2\right)}^{2} = 4$

This rule is particularly useful if you have more difficult problems such as an algebraic expression (with letters):
Consider: ${\left(\frac{x + 1}{4 x}\right)}^{2}$
You can now write:
${\left(\frac{x + 1}{4 x}\right)}^{2} = {\left(x + 1\right)}^{2} / {\left(4 x\right)}^{2} = \frac{{x}^{2} + 2 x + 1}{16 {x}^{2}}$

• The Quotient Rule for Exponents

Let me give you a basic explanation:

Lets take the example of
${4}^{36} / {4}^{21}$

The quotient rule states that for an expression like ${x}^{a} / {x}^{b} = {x}^{a - b}$

Now of course you question how to simplify expressions using this rule.

Now lets take such a eg.

Compute the following: $\frac{625 {x}^{23}}{25 {x}^{3}}$

this nothing but 25$\left({x}^{23 - 3}\right)$

So we are left with this final answer $25 {x}^{20}$