### Exponential Properties Involving Quotients

The rules for simplifying quotients of exponents are a lot like the rules for simplifying products.

Let’s look at what happens when we divide

You can see that when we divide two powers of

**Quotient Rule for Exponents:**

When we have expressions with more than one base, we apply the quotient rule separately for each base:

Now let’s see what happens if the exponent in the denominator is bigger than the exponent in the numerator. For example, what happens when we apply the quotient rule to

The quotient rule tells us to subtract the exponents. 4 minus 7 is -3, so our answer is

OR

Well, let’s look at what we get when we do the division longhand by writing each term in factored form:

Even when the exponent in the denominator is bigger than the exponent in the numerator, we can still subtract the powers. The

#### Simplifying Expressions

Simplify the following expressions, leaving all exponents positive.

a)

Subtract the exponent in the numerator from the exponent in the denominator and leave the

b)

Apply the rule to each variable separately:

### Examples

Simplify each of the following expressions using the quotient rule.

#### Example 1

#### Example 2

#### Example 3

c)

### Review

Evaluate the following expressions.

5652 6763 34310 22⋅3252 33⋅5237

Simplify the following expressions.

a3a2 x5x9 x6y2x2y5 6a32a2 15x55x 25yx620y5x2

### Review (Answers)

To view the Review answers, open this PDF file and look for section 8.3.