Indices & Surds: Definition of Surds

by Wonder Boy · Published October 8, 2021 · Updated October 8, 2021

Any root of a rational number, which cannot be further simplified to remove the root, it is called as Surd or Radical.

“Surds are irrational numbers.”

Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression,

For Example:
$\sqrt{3}$ , $\sqrt[5]{4}$ , $3 + \sqrt{3}$

Surds have infinite non-recurring decimals.
For Example:
$\sqrt{2}$ = 1.41421356237…
$\sqrt{3}$ = 1.73205080757…
$\sqrt{5}$ = 2.2360679775…
$\sqrt[5]{4}$ = 1.31950791077…
$3 + \sqrt{3}$ = 4.73205080757…

In the surd $\sqrt[n]{a}$ , a is called the radicand and n is called as the order of the surd.
$\sqrt{ }$ is the called as radical sign.

For $\sqrt[n]{a}$ to be considered as a surd:
1.) The radicand should be a positive rational number.
2.) n should be a natural number.
3.) $\sqrt[n]{a}$ should be an irrational number.

Types of Surds

1. Pure Surd: A surd having no rational factor except unity is called a pure surd or complete surd.
In pure surd, the whole of the rational number is under the radical sign and makes the radicand.

For Example:
$\sqrt{3}$ , $\sqrt[5]{4}$

2. Mixed Surd: A surd having factor which is a rational number except unity is called a mixed surd.

For Example:
$2\sqrt{3}$ , $\frac{4}{7}\sqrt[5]{4}$ , $-3\sqrt[5]{57}$

Every mixed surd can be expressed as a pure surd.
A pure surd can also be converted into a mixed surd but not always.

For Example:
Let’s convert a mixed surd $2\sqrt{3}$ to a pure surd.
$2\sqrt{3}$ = $\sqrt{2^2 \times 3}$ = $\sqrt{12}$

Let’s convert a pure surd $\sqrt{12}$ to a mixed surd.
$\sqrt{12}$ = $\sqrt{2^2 \times 3}$ = $2\sqrt{3}$

3. Monomial or Simple Surd:
A surd having a single term only is called a monomial or simple surd.

For Example:
$\sqrt{2}$ , $\sqrt{5}$ , $\sqrt[3]{7}$ , $5\sqrt[5]{12}$ are simple surds.

4. Compound Surds :
The algebraic sum of two or more simple surds or
the algebraic sum of a rational number and simple surds is called a Compound Surd.

For Example:
$3 + \sqrt{2}$ , $2 + \sqrt{5}$ are compound surds.
$\sqrt{7} + \sqrt{3}$ , $\sqrt{5} + \sqrt{2}$ are compound surds.
$3 + \sqrt[3]{2}$ , $2 + \sqrt[4]{5}$ are compound surds.
$\sqrt[4]{7} + \sqrt[5]{3}$ , $\sqrt[3]{5} + \sqrt[4]{2}$ are compound surds.
$3 + \sqrt[3]{2} + \sqrt[4]{5}$ are compound surds.

5. Binomial Surds:
A compound surd consisting of two surds is called a Binomial Surd i.e. the surds of the form $a\sqrt[n]{b} \pm c\sqrt[m]{d}$ or $a\pm b\sqrt[n]{c}$ are called binomial surds.

For Example:
$\sqrt[5]{7} + \sqrt[4]{3}$ , $\sqrt[3]{5} + \sqrt[5]{2}$ are Binomial surds.

6. Binomial Quadratic Surds:
Binomial surds consisting of pure (or simple) surds of order two i.e., the surds of the form $a\sqrt{b} \pm c\sqrt{d}$ or $a\pm b\sqrt{c}$ are called binomial quadratic surds.

For Example:
$\sqrt{7} + \sqrt{3}$ , $\sqrt{5} + \sqrt{2}$ are Binomial Quadratic surds.

7. Similar Surds:
Surds which have the same surd in their simplest form are called as Similar Surds.

For Example:
$\sqrt{98}$ and $\sqrt{50}$ are similar surds.
Simplest form of $\sqrt{98}$ is $7\sqrt{2}$ .
Simplest form of $\sqrt{50}$ is $5\sqrt{2}$ .

$\sqrt{98}$ and $\sqrt{50}$ have the same surd $\sqrt{2}$ in the simplest form.

8. Conjugate Surds or Complementary Surds:
The sum and difference of two binomial quadratic surds are said to be conjugate surds to each other. They differ only in sign which connects the two terms.
Conjugate surds are also known as complementary surds.

For Example:
1. The sum and the difference of two simple quadratic surds $5\sqrt{3}$ and $\sqrt{5}$ are $5\sqrt{3} + \sqrt{5}$ and $5\sqrt{3} - \sqrt{5}$ respectively.
Therefore, two surds $5\sqrt{3} + \sqrt{5}$ and $5\sqrt{3} - \sqrt{5}$ are conjugate to each other.

The product of binomial quadratic surd and its conjugate will always result in a rational number.

For Example:
The product of $2 + \sqrt{3}$ and $2 - \sqrt{3}$ = 4-3 = 1 (which is a rational number.)

Rationalisation of Surds
If product of two surds results into a rational number, then each surd is said to be rationalised when multiplied by each other..
And each surd is said to be rationalising factor of each other.

For Example:
1. Product of $2\sqrt{3}$ and $\sqrt{3}$ = 6. Hence, 2 $\sqrt{3}$ and $\sqrt{3}$ can rationalise each other.

2. Product of $2 + \sqrt{3}$ and $2 - \sqrt{3}$ = $2^2 - (\sqrt{3})^2$ = 4-3 =1. Hence, $2 + \sqrt{3}$ and $2 - \sqrt{3}$ can rationalise each other.

Laws of Surds:
The manipulation of indices can be a powerful tool to simplify an expression. There are various laws of indices which govern the whole operation.

1. $(\sqrt[n]{a})^n = a$

For Example:
$(\sqrt[5]{7})^5 = 7$

2. $\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{ab}$

For Example:
$\sqrt[5]{3} \sqrt[5]{7} = \sqrt[5]{21}$

3. $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$

For Example:
$\frac{\sqrt[5]{3}}{\sqrt[5]{7}} = \sqrt[5]{\frac{3}{7}}$

4. $\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} = \sqrt[n]{\sqrt[m]{a}}$

For Example:
$\sqrt[3]{\sqrt[2]{5}} = \sqrt[6]{5} = \sqrt[2]{\sqrt[3]{5}}$

5. $(\sqrt[n]{a})^m = \sqrt[n]{a^m}$

For Example:
$(\sqrt[5]{3})^2 = \sqrt[5]{3^2}$

Operational Laws:

1. Surds cannot be added
$\sqrt{a} + \sqrt{b} \neq \sqrt{a + b}$

But, we can add similar surds.
$x\sqrt{a} + y\sqrt{a} = (x + y)\sqrt{a}$

2. Surds cannot be Subtracted
$\sqrt{a} - \sqrt{b} \neq \sqrt{a - b}$

But, we can subtract similar surds.
$x\sqrt{a} - y\sqrt{a} = (x - y)\sqrt{a}$

3. Surds can be multiplied
$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

4. Surds can be divided
$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

5. Surds can also be expressed in exponential form (indices form)
$\sqrt[n]{a} = a^{1/n}$