Lesson

We want to look at expressions of the form $x^{\frac{m}{n}}$`x``m``n`. Let's make a connection to previous knowledge and start with a particular example of $x^{\frac{1}{2}}$`x`12.

Consider the process below using the index laws we have previously looked at:

$x^{\frac{2}{2}}$x22 |
$=$= | $x$x |

$\left(x^{\frac{1}{2}}\right)^2$(x12)2 |
$=$= | $x$x |

$\left(x^{\frac{1}{2}}\right)^2$(x12)2 |
$=$= | $\left(\sqrt{x}\right)^2$(√x)2 |

$x^{\frac{1}{2}}$x12 |
$=$= | $\sqrt{x}$√x |

Fractional indices

$x^{\frac{1}{n}}=\sqrt[n]{x}$`x`1`n`=^{n}√`x`

You may also see questions with more complicated fractional indices, such as $x^{\frac{3}{2}}$`x`32. We could express this as a power of a power, using the rule $\left(x^a\right)^b=x^{ab}$(`x``a`)`b`=`x``a``b`, as follows:

$x^{\frac{3}{2}}=\left(x^3\right)^{\frac{1}{2}}$`x`32=(`x`3)12$=$=$\sqrt{x^3}$√`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:

Fractional indices

$x^{\frac{m}{n}}=\sqrt[n]{x^m}$`x``m``n`=^{n}√`x``m`

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

Consider the following.

Rewrite $x^{\frac{1}{3}}$

`x`13 in surd form.Evaluate $\sqrt[3]{x}$

^{3}√`x`for when $x=8$`x`=8.

Write the following with a fractional index:

$\sqrt[8]{23}$Simplify the following, leaving your answer in index form:

$5^{-\frac{2}{9}}\times5^{\frac{4}{9}}$5−29×549.

Write the following as a simplified fraction:

$\left(\frac{16}{9}\right)^{-\frac{1}{2}}$(169)−12.

Define rational and irrational numbers and perform operations with surds and fractional indices.