## Combining Properties to Simplify Expressions

### Learning Outcomes

• Simplify quotients that require a combination of the properties of exponents

We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

### Summary of Exponent Properties

If $a,b$ are real numbers and $m,n$ are whole numbers, then

$\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left({a}^{m}\right)}^{n}={a}^{m\cdot n}\hfill \\ \mathbf{\text{Product to a Power Property}}\hfill & & & {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},a\ne 0,m>n\hfill \\ & & & \frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},a\ne 0,n>m\hfill \\ \mathbf{\text{Zero Exponent Definition}}\hfill & & & {a}^{0}=1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & {\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne 0\hfill \end{array}$

### example

Simplify: ${\Large\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}}$.

Solution

 ${\Large\frac{{\left({x}^{2}\right)}^{3}}{{x}^{5}}}$ Multiply the exponents in the numerator, using the Power Property. ${\Large\frac{{x}^{6}}{{x}^{5}}}$ Subtract the exponents. $x$

### example

Simplify: ${\Large\frac{{m}^{8}}{{\left({m}^{2}\right)}^{4}}}$

### example

Simplify: ${\left({\Large\frac{{x}^{7}}{{x}^{3}}}\right)}^{2}$

### example

Simplify: ${\left({\Large\frac{{p}^{2}}{{q}^{5}}}\right)}^{3}$

### example

Simplify: ${\Large{\left(\frac{2{x}^{3}}{3y}\right)}}^{4}$

### example

Simplify: ${\Large\frac{{\left({y}^{2}\right)}^{3}{\left({y}^{2}\right)}^{4}}{{\left({y}^{5}\right)}^{4}}}$

### try it

For more similar examples, watch the following video.