1.08 Fractional powers
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:
$x^{\frac{1}{n}}=\sqrt[n]{x}$x1n=n√x
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:
$x^{\frac{m}{n}}=\sqrt[n]{x^m}$xmn=n√xm
$x^{\frac{m}{n}}=\left(\sqrt[n]{x}\right)^m$xmn=(n√x)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 |
Now let's start with the powers:
| $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}$3√x2 with a fractional exponent.
Think: We know that $x^{\frac{m}{n}}=\sqrt[n]{x^m}$xmn=n√xm.
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}}$4−34×414.
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}$5√x7 in exponential form.
Question 5
Evaluate $4^{\frac{3}{2}}$432.
Question 6
Fully simplify $\left(\frac{25}{36}\right)^{\frac{1}{2}}$(2536)12.