# 4.02 Fractional powers

Lesson

We want to look at expressions of the form $x^{\frac{m}{n}}$xmn. Let's make a connection to previous knowledge and start with a particular example of $x^{\frac{1}{2}}$x12.

Consider the process below:

 $x^{\frac{2}{2}}$x22​ $=$= $x$x $\left(x^{\frac{1}{2}}\right)^2$(x12​)2 $=$= $x$x Product of powers property $\left(x^{\frac{1}{2}}\right)^2$(x12​)2 $=$= $\left(\sqrt{x}\right)^2$(√x)2 Rewrite $x$x using the definition of square root $x^{\frac{1}{2}}$x12​ $=$= $\sqrt{x}$√x Taking the square root both sides

In general, we get that:

Fractional exponents

$x^{\frac{1}{n}}=\sqrt[n]{x}$x1n=nx

### More fractional exponents

You may also see questions with more complicated fractional exponents, such as $x^{\frac{3}{2}}$x32. We could express this as a power of a power, $\left(x^3\right)^{\frac{1}{2}}$(x3)12. As such, the numerator in the fractional exponent can be expressed as a power and the denominator in the fractional exponent can be expressed as a root.

More generally, this rule states:

Fractional exponents

$x^{\frac{m}{n}}=\sqrt[n]{x^m}$xmn=nxm

$x^{\frac{m}{n}}=\left(\sqrt[n]{x}\right)^m$xmn=(nx)m

When solving problems with fractional exponents, it doesn't matter whether you start with the powers or the roots (although you might find it easier to do it one way than the other).

For example, let's look at $16^{\frac{3}{2}}$1632

Starting with the root:

 $16^{\frac{3}{2}}$1632​ $=$= $\left(\sqrt{16}\right)^3$(√16)3 $=$= $4^3$43 $=$= $64$64

 $16^{\frac{3}{2}}$1632​ $=$= $\sqrt{16^3}$√163 $=$= $\sqrt{4096}$√4096 $=$= $64$64

You can see that we get the same answer both ways, but the second approach led to working with some much larger numbers

#### Worked examples

##### Question 1

Rewrite: $\sqrt[3]{x^2}$3x2 with a fractional exponent.

Think: We know that $x^{\frac{m}{n}}=\sqrt[n]{x^m}$xmn=nxm.

Do:

 $\sqrt[3]{x^2}$3√x2 $=$= $\left(x^2\right)^{\frac{1}{3}}$(x2)13​ $=$= $x^{\frac{2}{3}}$x23​
##### Question 2

Evaluate: $121^{\frac{1}{2}}$12112.

Think: A fractional exponent of $\frac{1}{2}$12 is the same as finding the square root.

Do:

 $121^{\frac{1}{2}}$12112​ $=$= $\sqrt{121}$√121 $=$= $11$11
##### Question 3

Evaluate: $4^{-\frac{3}{4}}\times4^{\frac{1}{4}}$434×414.

Think: We should use our laws of exponents to simplify first and then use the definition of rational exponents.

Do:

 $4^{-\frac{3}{4}}\times4^{\frac{1}{4}}$4−34​×414​ $=$= $4^{-\frac{3}{4}+\frac{1}{4}}$4−34​+14​ $=$= $4^{-\frac{2}{4}}$4−24​ $=$= $4^{-\frac{1}{2}}$4−12​ $=$= $\frac{1}{4^{\frac{1}{2}}}$1412​​ $=$= $\frac{1}{\sqrt{4}}$1√4​ $=$= $\frac{1}{2}$12​

#### Practice questions

##### Question 4

Express $\sqrt[5]{x^7}$5x7 in exponential form.

##### Question 5

Evaluate $4^{\frac{3}{2}}$432.

##### Question 6

Evaluate $\left(\frac{25}{36}\right)^{\frac{1}{2}}$(2536)12.

### Outcomes

#### MGSE9-12.N.RN.1

Explain how the meaning of rational exponents follows from extending the properties of integer exponents to rational numbers, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want [5^(1/3)]^3 = 5^[(1/3) x 3] to hold, so [5^(1/3)]^3 must equal 5.

#### MGSE9-12.N.RN.2

Rewrite expressions involving radicals and rational exponents using the properties of exponents.