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# Fractional Exponents

## Relate fractional exponents to nth roots

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Fractional Exponents

### Fractional Exponents

In a previous section we looked at the quotient rule for exponents: (xnxm=x(nm))\begin{align*}\left ( \frac{x^n}{x^m} = x^{(n-m)}\right )\end{align*}. Consider what happens when n=m\begin{align*}n = m\end{align*}.

For example, what happens when we divide x4\begin{align*}x^4\end{align*} by x4\begin{align*}x^4\end{align*}? Applying the quotient rule tells us that x4x4=x(44)=x0\begin{align*}\frac{x^4}{x^4} = x^{(4-4)}=x^0\end{align*}—so what does that zero mean?

Well, we first discovered the quotient rule by considering how the factors of x\begin{align*}x\end{align*} cancel in such a fraction. Let’s do that again with our example of x4\begin{align*}x^4\end{align*} divided by x4\begin{align*}x^4\end{align*}:

x4x4=xxxxxxxx=1\begin{align*}\frac{x^4}{x^4} = \frac{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot \cancel{x}}{\cancel{x} \cdot \cancel{x} \cdot \cancel{x} \cdot \cancel{x}} = 1\end{align*}

So x0=1!\begin{align*}x^0=1 !\end{align*} You can see that this works for any value of the exponent, not just 4:

xnxn=x(nn)=x0\begin{align*}\frac{x^n}{x^n} = x^{(n-n)} = x^0\end{align*}

Since there is the same number of x\begin{align*}x\end{align*}’s in the numerator as in the denominator, they cancel each other out and we get x0=1\begin{align*}x^0 = 1\end{align*}. This rule applies for all expressions:

Zero Rule for Exponents: x0=1\begin{align*}x^0=1\end{align*}, where x0\begin{align*}x \neq 0\end{align*}

#### 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 912\begin{align*}9^{\frac{1}{2}}\end{align*}—how can we relate this expression to one that we already know how to work with? For example, how could we turn it into an expression that doesn’t have any fractional exponents?

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 912\begin{align*}9^ \frac{1}{2}\end{align*} to the power of 2, we get (912)2=9212=91=9\begin{align*}\left(9^\frac{1}{2}\right)^2=9^{2 \cdot \frac{1}{2}}=9^1=9\end{align*}

So if 912\begin{align*}9^{\frac{1}{2}}\end{align*} squared equals 9, what does 912\begin{align*}9^{\frac{1}{2}}\end{align*} itself equal? Well, 3 is the number whose square is 9 (that is, it’s the square root of 9), so 912\begin{align*}9^{\frac{1}{2}}\end{align*} must equal 3. And that’s true for all numbers and variables: a number raised to the power of 12\begin{align*}\frac{1}{2}\end{align*} is just the square root of the number. We can write that as x=x12\begin{align*}\sqrt{x}=x^{\frac{1}{2}}\end{align*}, and then we can see that’s true because (x)2=x\begin{align*}\left( \sqrt{x} \right)^2=x\end{align*} just as (x12)2=x\begin{align*}\left( x^{\frac{1}{2}} \right)^2=x\end{align*}.

Similarly, a number to the power of 13\begin{align*}\frac{1}{3}\end{align*} is just the cube root of the number, and so on. In general, x1n=xn\begin{align*}x^{\frac{1}{n}}=\sqrt[n]{x}\end{align*}. And when we raise a number to a power and then take the root of it, we still get a fractional exponent; for example, x43=(x4)13=x43\begin{align*}\sqrt[3]{x^4}=\left(x^4\right)^{\frac{1}{3}}=x^{\frac{4}{3}}\end{align*}. In general, the rule is as follows:

Rule for Fractional Exponents: anm=anm\begin{align*}\sqrt[m]{a^n}=a^{\frac{n}{m}}\end{align*} and (am)n=anm\begin{align*}\left( \sqrt[m]{a}\right)^n=a^{\frac{n}{m}}\end{align*}

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) a12a13\begin{align*}a^{\frac{1}{2}} \cdot a^{\frac{1}{3}}\end{align*}

Apply the product rule: a12a13=a12+13=a56\begin{align*}a^\frac{1}{2} \cdot a^\frac{1}{3}=a^{\frac{1}{2}+\frac{1}{3}}=a^{\frac{5}{6}}\end{align*}

b) (a13)2\begin{align*}\left(a^{\frac{1}{3}}\right)^2\end{align*}

Apply the power rule: (a13)2=a23\begin{align*}\left(a^{\frac{1}{3}}\right)^2=a^{\frac{2}{3}}\end{align*}

c) a52a12\begin{align*}\frac{a^{\frac{5}{2}}}{a^{\frac{1}{2}}}\end{align*}

Apply the quotient rule: a52a12=a5212=a42=a2\begin{align*}\frac{a^{\frac{5}{2}}}{a^{\frac{1}{2}}}=a^{\frac{5}{2}-\frac{1}{2}}=a^{\frac{4}{2}}=a^2\end{align*}

d) (x2y3)13\begin{align*}\left(\frac{x^2}{y^3}\right)^{\frac{1}{3}}\end{align*}

Apply the power rule for quotients: (x2y3)13=x23y\begin{align*}\left(\frac{x^2}{y^3}\right)^{\frac{1}{3}}=\frac{x^{\frac{2}{3}}}{y}\end{align*}

### Examples

Simplify the following expressions.

#### Example 1

(x25)5\begin{align*}\left(x^{\frac{2}{5}}\right)^5\end{align*}

Apply the power rule: (x25)5=x255=x2\begin{align*}\left(x^{\frac{2}{5}}\right)^5=x^{\frac{2}{5}\cdot 5}=x^2\end{align*}

#### Example 2

y34y18\begin{align*}\frac{y^{\frac{3}{4}}}{y^{\frac{1}{8}}}\end{align*}

Apply the quotient rule: y34y18=y3418=y6818=y58\begin{align*}\frac{y^{\frac{3}{4}}}{y^{\frac{1}{8}}}=y^{\frac{3}{4}-\frac{1}{8}}=y^{\frac{6}{8}-\frac{1}{8}}=y^{\frac{5}{8}}\end{align*}

#### Example 3

(x2ay4b)12\begin{align*}\left(\frac{x^{2a}}{y^{4b}}\right)^{\frac{1}{2}}\end{align*}

Apply the power rule for quotients: (x2ay4b)12=x2a12y4b12=xay2b\begin{align*}\left(\frac{x^{2a}}{y^{4b}}\right)^{\frac{1}{2}}=\frac{x^{2a\cdot \frac{1}{2}}}{y^{4b\cdot \frac{1}{2}}}=\frac{x^{a}}{y^{2b}}\end{align*}

### Review

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

1. (x12y23)(x2y13)\begin{align*}(x^\frac{1}{2} y^\frac{-2}{3})(x^2y^\frac{1}{3})\end{align*}
2. x3x3\begin{align*}x^{-3} \cdot x^3\end{align*}
3. y5y5\begin{align*}y^{5} \cdot y^{-5}\end{align*}
4. x2y3x4y2x2\begin{align*}\frac{x^2y^{-3}}{x^{-4}y^{-2}}\cdot x^2\end{align*}

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

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

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