# Surds

Go back to  'Irrational Numbers'

## Introduction:

Around 820 BC, Muhammad ibn Musa al-Khwarizmi (a Persian scholar) called rational numbers "audible" and irrational numbers 'inaudible". This later led to the Arabic name asamm (deaf, dumb) for the irrational numbers being translated as surdus (deaf or mute) into Latin.

✍Note: For understanding surds, we should have the basic knowledge of rational exponents. A surd is an advanced form of the exponent.

## What are Surds?

Suppose that $$b$$ is a rational number which is not a perfect $${n^{th}}$$ power of another rational number (where $$n$$ is a natural number). Then, the irrational number $${b^{\left( {\frac{1}{n}} \right)}}$$ or $$\sqrt[n]{b}$$ is an example of a simple surd.

Now, suppose that $$c$$ is another rational number. Then the irrational number $$c + {b^{\left( {\frac{1}{n}} \right)}}$$ is an example of a mixed surd.

We can also have compound surds, which are formed by the sum/difference of two or more surds.

We see that surds are irrational numbers that can be expressed using the roots of rational numbers, in the manner we have described above.

✍Note: Every surd is an irrational number, but every irrational number is not a surd.

Here are some examples of irrational numbers classified as surds and non-surds:

Surds Non-surds
$\sqrt 3$ $\pi$
$1 + \sqrt 3$ $\sqrt[3]{{1 + \sqrt 3 }}$
$\sqrt 2 + \sqrt[4]{3} + 1$ $\sqrt[5]{{\sqrt 2 + 1}}$
$\sqrt 8 - \sqrt[6]{{11}} + \sqrt[7]{9} - \sqrt[7]{{13}}$ $\sqrt \pi + \sqrt {2 + } \sqrt[3]{3}$

✍Note: Keeping numbers in surd form avoids rounding errors, which may affect a calculation requiring precise results.

## Solved Examples:

Example 1: Classify the following as surds and non-surds?

$\sqrt {15} ,{\text{ }}\sqrt {16} ,{\text{ }}\sqrt {17} ,{\text{ }}\sqrt {18}$

Solution: We know that 15, 17, and 18 are not perfect squares but 16 is a perfect square of 4.

So,

$\boxed{\sqrt {15} ,\sqrt {17} ,\sqrt {18} {\text{ are surds}}}$

$\boxed{\sqrt {16} {\text{ is non - surd}}}$

Example 2: Which of the following are surds?

$\sqrt {729} ,{\text{ }}\sqrt[3]{{729}},{\text{ }}\sqrt[4]{{729}},{\text{ }}\sqrt[5]{{729}},{\text{ }}\sqrt[6]{{729}}$

Solution: Let's try to simplify each of the above,

(1) $$\sqrt {729}$$

\begin{align} \sqrt {729} &= \sqrt {3 \times 3 \times 3 \times 3 \times 3 \times 3} \hfill \\ &= 3 \times 3 \times 3 \hfill \\ &= 27...({\text{non - surd}}) \hfill \\ \end{align}

(2) $$\sqrt[3]{{729}}$$

\begin{align} \sqrt[3]{{729}} &= \sqrt[3]{{3 \times 3 \times 3 \times 3 \times 3 \times 3}} \hfill \\ &= 3 \times 3 \hfill \\ &= 9...({\text{non - surd}}) \hfill \\ \end{align}

(3) $$\sqrt[4]{{729}}$$

\begin{align} \sqrt[4]{{729}} &= \sqrt[4]{{3 \times 3 \times 3 \times 3 \times 3 \times 3}} \hfill \\ &= 3 \times \sqrt[4]{9}...({\text{surd}}) \hfill \\ \end{align}

(4) $$\sqrt[5]{{729}}$$

\begin{align} \sqrt[5]{{729}} &= \sqrt[5]{{3 \times 3 \times 3 \times 3 \times 3 \times 3}} \hfill \\ &= 3 \times \sqrt[5]{3}...({\text{surd}}) \hfill \\ \end{align}

(5) $$\sqrt[6]{{729}}$$

\begin{align} \sqrt[6]{{729}} &= \sqrt[6]{{3 \times 3 \times 3 \times 3 \times 3 \times 3}} \hfill \\ &= 3...({\text{non - surd}}) \hfill \\ \end{align}

Thus,

$\boxed{\sqrt[4]{{729}}{\text{ and }}\sqrt[5]{{729}}{\text{ are surds}}{\text{.}}}$

Challenge: Which of the following are non-surds?

$\sqrt {1024} ,{\text{ }}\sqrt[3]{{1024}},{\text{ }}\sqrt[4]{{1024}},{\text{ }}\sqrt[5]{{1024}}$

⚡Tip: Use a similar approach as in example-2.