2.06 Fractional indices

We want to look at expressions of the form x^{\frac{m}{n}}. Let's make a connection to previous knowledge and start with a particular example of x^{\frac{1}{2}}. Consider the process below using the index laws we have previously looked at:

You may also see questions with more complicated fractional indices, such as x^{\frac{3}{2}}. We could express this as a power of a power, using the rule\left(x^{a}\right)^{b}=x^{ab}, as follows: x^{\frac{3}{2}}=\left(x^{3}\right)^{\frac{1}{2}}=\sqrt{x^{3}}

As such, the numerator in the fractional index can be expressed as a power and the denominator in the fractional index can be expressed as a root.

More generally, this rule states: x^{\frac{m}{n}}=\sqrt[n]{x^{m}}=\left(\sqrt[n]{x}\right)^{m}

When solving problems with fractional indices, 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}}:

Starting with the root:

Now let's start with the powers:

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

Examples