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10&10a

2.06 Fractional indices

Lesson

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:

\displaystyle x^{\frac{2}{2}}\displaystyle =\displaystyle x
\displaystyle \left(x^{\frac{1}{2}}\right)^{2}\displaystyle =\displaystyle x
\displaystyle \left(x^{\frac{1}{2}}\right)^{2}\displaystyle =\displaystyle \left(\sqrt{x}\right)^{2}
\displaystyle x^{\frac{1}{2}}\displaystyle =\displaystyle \sqrt{x}

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}}

The image shows a base of x and exponent of m over n.

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:

\displaystyle 16^{\frac{3}{2}}\displaystyle =\displaystyle \left(\sqrt{16}\right)^{3}
\displaystyle =\displaystyle 4^{3}
\displaystyle =\displaystyle 64

Now let's start with the powers:

\displaystyle 16^{\frac{3}{2}}\displaystyle =\displaystyle \left(\sqrt{16^{3}}\right)
\displaystyle =\displaystyle \sqrt{4096}
\displaystyle =\displaystyle 64

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

Examples

Example 1

Consider the following.

a

Rewrite x^{\frac{1}{3}} in surd form.

Worked Solution
Create a strategy

Use the rule A^{\frac{1}{n}}=\sqrt[n]{A}.

Apply the idea

x^{\frac{1}{3}}=\sqrt[3]{x}

b

Evaluate \sqrt[3]{x} for when x=8.

Worked Solution
Create a strategy

Find the cube root of 8.

Apply the idea
\displaystyle \sqrt[3]{x}\displaystyle =\displaystyle \sqrt[3]{8}Substitute x=8
\displaystyle =\displaystyle 2Evaluate

Example 2

Write the following with a fractional index: \sqrt[7]{72}

Worked Solution
Create a strategy

Use the rule \sqrt[n]{A}=A^{\frac{1}{n}}.

Apply the idea

\sqrt[7]{72}=72^{\frac{1}{7}}

Example 3

Write the following as a simplified fraction: \left(\dfrac{16}{9}\right)^{-\frac{1}{2}}

Worked Solution
Create a strategy

Use the rule: \left(\dfrac{A}{B}\right)^{-\frac{1}{n}}=\left(\dfrac{B}{A}\right)^{\frac{1}{n}}

Apply the idea
\displaystyle \left(\dfrac{16}{9}\right)^{-\frac{1}{2}}\displaystyle =\displaystyle \left(\dfrac{9}{16}\right)^{\frac{1}{2}}Apply the rule
\displaystyle =\displaystyle \dfrac{\sqrt{9}}{\sqrt{16}}Square root the numerator and denominator
\displaystyle =\displaystyle \dfrac{3}{4}Evaluate the square roots
Idea summary

Fractional index laws:

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

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

Outcomes

ACMNA264 (10a)

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

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