Mathematics » Exponents and Surds » Rational Exponents And Surds

# Rational Exponents and Surds

## Rational Exponents and Surds

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:

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

## 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*}