## Rational Exponents and Surds

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The laws of exponents can also be extended to include the rational numbers.A rational number is any number that can be written as a fraction with an integer in the numerator and in the denominator.We also have the following definitions for working with rational exponents.

- If \(r^n = a\), then \(r = \sqrt[n]{a} \quad (n \geq 2)\)
- \(a^{\frac{1}{n}} = \sqrt[n]{a}\)
- \(a^{-\cfrac{1}{n}} = (a^{-1})^{\frac{1}{n}} = \sqrt[n]{\cfrac{1}{a}}\)
- \(a^{\frac{m}{n}} = (a^{m})^{\frac{1}{n}} = \sqrt[n]{a^m}\)

where \(a > 0\), \(r > 0\) and \(m,n \in \mathbb{Z}\), \(n \ne 0\).

For \(\sqrt{25} = 5\), we say that \(\text{5}\) is the square root of \(\text{25}\) and for \(\sqrt[3]{8} = 2\), we say that \(\text{2}\) is the cube root of \(\text{8}\). For \(\sqrt[5]{32} = 2\), we say that \(\text{2}\) is the fifth root of \(\text{32}\).

When dealing with exponents, a root refers to a number that is repeatedly multiplied by itself a certain number of times to get another number.A radical refers to a number written as shown below.

The radical symbol and degree show which root is being determined. The radicand is the number under the radical symbol.

If \(n\) is an even natural number, then the radicand must be positive, otherwise the roots are not real. For example, \(\sqrt[4]{16} = 2\) since \(2 \times 2 \times 2 \times 2 = 16\), but the roots of \(\sqrt[4]{-16}\) are not real since \((-2) \times (-2) \times (-2) \times (-2) \ne -16\).

If \(n\) is an odd natural number, then the radicand can be positive or negative. For example, \(\sqrt[3]{27} = 3\) since \(3 \times 3 \times 3 = 27\) and we can also determine \(\sqrt[3]{-27} = -3\) since \((-3) \times (-3) \times (-3) = -27\).

It is also possible for there to be more than one \(n^{\text{th}}\) root of a number. For example, \((-2)^2 = 4\) and \(2^2 = 4\), so both \(-\text{2}\) and \(\text{2}\) are square roots of \(\text{4}\).

A surd is a radical which results in an irrational number. Irrational numbers are numbers that cannot be written as a fraction with the numerator and the denominator as integers. For example, \(\sqrt{12}\), \(\sqrt[3]{\text{100}}\), \(\sqrt[5]{25}\) are surds.

## Example

### Question

Write each of the following as a radical and simplify where possible:

- \(18^\frac{1}{2}\)
- \((-\text{125})^{-\cfrac{1}{3}}\)
- \(4^\frac{3}{2}\)
- \((-81)^\frac{1}{2}\)
- \((\text{0.008})^\frac{1}{3}\)

- \(18^\frac{1}{2} = \sqrt{18}\)
- \((-\text{125})^{-\cfrac{1}{3}} = \sqrt[3]{(-\text{125})^{-1}} = \sqrt[3]{\dfrac{1}{-\text{125}}} = \sqrt[3]{\dfrac{1}{(-5)^3}} = -\dfrac{1}{5}\)
- \(4^\frac{3}{2} = (4^3)^{\frac{1}{2}} = \sqrt{4^3} = \sqrt{64} = 8\)
- \((-81)^\frac{1}{2} = \sqrt{-81} =\) not real
- \((\text{0.008})^\frac{1}{3} = \sqrt[3]{\dfrac{8}{\text{1 000}}} = \sqrt[3]{\dfrac{2^3}{10^3}} = \cfrac{2}{10} = \dfrac{1}{5}\)

## Example

### Question

Simplify without using a calculator:

\[\left(\cfrac{5}{4^{-1}-9^{-1}}\right)^{\frac{1}{2}}\]

### Write the fraction with positive exponents in the denominator

\[( \dfrac{5}{\cfrac{1}{4} – \cfrac{1}{9}} )^{\frac{1}{2}}\]

### Simplify the denominator

\begin{align*} &= ( \dfrac{5}{\cfrac{9-4}{36}} )^{\frac{1}{2}} \\ &= ( \dfrac{5}{\cfrac{5}{36}} )^{\frac{1}{2}} \\ &= ( 5 \div \cfrac{5}{36} )^{\frac{1}{2}} \\ &= ( 5 \times \cfrac{36}{5} )^{\frac{1}{2}} \\ &= (36)^{\frac{1}{2}} \end{align*}

### Take the square root

\begin{align*} &= \sqrt{36}\\ &= 6 \end{align*}