### Fractional Exponents

In a previous section we looked at the quotient rule for exponents:

For example, what happens when we divide

Well, we first discovered the quotient rule by considering how the factors of

So

Since there is the same number of

**Zero Rule for Exponents:**

**Simplify Expressions With Fractional Exponents**

So far we’ve only looked at expressions where the exponents are positive and negative integers. The rules we’ve learned work exactly the same if the powers are fractions or irrational numbers—but what does a fractional exponent even mean? Let’s see if we can figure that out by using the rules we already know.

Suppose we have an expression like

Well, the power rule tells us that if we raise an exponential expression to a power, we can multiply the exponents.

For example, if we raise

So if *squared* equals 9, what does

Similarly, a number to the power of

**Rule for Fractional Exponents:**

We’ll examine roots and radicals in detail in a later chapter. In this section, we’ll focus on how exponent rules apply to fractional exponents.

#### Simplifying Expressions

a)

Apply the product rule:

b)

Apply the power rule:

c)

Apply the quotient rule:

d)

Apply the power rule for quotients:

### Examples

Simplify the following expressions.

**Example 1**

Apply the power rule:

#### Example 2

Apply the quotient rule:

#### Example 3

Apply the power rule for quotients:

### Review

Simplify the following expressions in such a way that there aren't any negative exponents in the answer.

(x12y−23)(x2y13) x−3⋅x3 y5⋅y−5 x2y−3x−4y−2⋅x2

Simplify the following expressions in such a way that there aren't any fractions in the answer.

x12y52 (ab)3/4 - \begin{align*}\frac{3x^2y^\frac{3}{2}}{xy^\frac{1}{2}}\end{align*}
- \begin{align*}\frac{x^\frac{1}{2}y^\frac{5}{2}}{x^\frac{3}{2} y^\frac{3}{2}}\end{align*}
- \begin{align*}\left(\frac{a^2b^{\frac{1}{3}}}{a^3b}\right)^\frac{1}{2}\end{align*}
- \begin{align*}\left(\frac{a^\frac{1}{2}b}{ab^\frac{1}{4}}\right)^2\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 8.6.