## Simplifying Complex Expressions I

### Learning Outcomes

• Simplify complex expressions using a combination of exponent rules
• Simplify quotients that require a combination of the properties of exponents

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference as we will be using them here to simplify various exponential expressions.

### Summary of Exponent Properties

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

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

### Expressions with negative exponents

The following examples involve simplifying expressions with negative exponents.

### example

Simplify:

1. ${x}^{-4}\cdot {x}^{6}$
2. ${y}^{-6}\cdot {y}^{4}$
3. ${z}^{-5}\cdot {z}^{-3}$

Solution

 1. ${x}^{-4}\cdot {x}^{6}$ Use the Product Property, ${a}^{m}\cdot {a}^{n}={a}^{m+n}$. ${x}^{-4+6}$ Simplify. ${x}^{2}$
 2. ${y}^{-6}\cdot {y}^{4}$ The bases are the same, so add the exponents. ${y}^{-6+4}$ Simplify. ${y}^{-2}$ Use the definition of a negative exponent, ${a}^{-n}={\Large\frac{1}{{a}^{n}}}$. ${\Large\frac{1}{{y}^{2}}}$
 3. ${z}^{-5}\cdot {z}^{-3}$ The bases are the same, so add the exponents. ${z}^{-5 - 3}$ Simplify. ${z}^{-8}$ Use the definition of a negative exponent, ${a}^{-n}={\Large\frac{1}{{a}^{n}}}$. ${\Large\frac{1}{{z}^{8}}}$

### try it

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.

### example

Simplify: $\left({m}^{4}{n}^{-3}\right)\left({m}^{-5}{n}^{-2}\right)$

### try it

If we multipy two expressions with numerical coefficients, we multiply the coefficients together.

### example

Simplify: $\left(2{x}^{-6}{y}^{8}\right)\left(-5{x}^{5}{y}^{-3}\right)$

### try it

In the next two examples, we’ll use the Power Property and the Product to a Power Property to simplify expressions with negative exponents.

### example

Simplify: ${\left({k}^{3}\right)}^{-2}$.

### example

Simplify: ${\left(5{x}^{-3}\right)}^{2}$

### try it

In the following video we show another example of how to simplify a product that contains negative exponents.

The following examples involve solving exponential expressions with quotients.

### 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.

To conclude this section, we will simplify quotient expressions with a negative exponent.

### example

Simplify: ${\Large\frac{{r}^{5}}{{r}^{-4}}}$.

### try it

In the next video we share more examples of simplifying a quotient with negative exponents.