In Mathematics, surds are the values in square root that cannot be further simplified into whole numbers or integers. Surds are irrational numbers. The examples of surds are √2, √3, √5, etc., as these values cannot be further simplified. If we further simply them, we get decimal values, such as:

√2  = 1.4142135…

√3 = 1.7320508…

√5 = 2.2360679…

Surds Definition

Surds are the square roots  (√) of numbers that cannot be simplified into a whole or rational number. It cannot be accurately represented in a fraction. In other words, a surd is a root of the whole number that has an irrational value. Consider an example, √2 ≈ 1.414213. It is more accurate if we leave it as a surd √2.

Types of Surds

The different types of surds are as follows:

• Simple Surds – A surd that has only one term is called a simple surd. Example: √2, √5, …
• Pure Surds – Surds which are completely irrational. Example: √3
• Similar Surds – The surds having the same common surds factor
• Mixed Surds – Surds that are not completely irrational and can be expressed as a product of a rational number and an irrational number
• Compound Surds – An expression which is the addition or subtraction of two or more surds
• Binomial Surds –  A surd that is made of two other surds

Six Rules for Surds

Rule 1:

\(\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}\)

Example:

To simplify √18

18 = 9 x 2 = 3² x 2, since 9 is the greatest perfect square factor of 18.

Therefore, √18 = √(3² x 2)

= √3² x √2

= 3 √2

Rule 2:

\(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\)

Example:

√(12 / 121) = √12 / √121

=√(2² x 3) / 11

=√2² x √3 / 11

= 2√3 / 11

Rule 3:

\(\frac{b}{\sqrt{a}}=\frac{b}{\sqrt{a}}\times \frac{\sqrt{a}}{\sqrt{a}}=b\frac{\sqrt{a}}{a}\)

You can rationalize the denominator by multiplying the numerator and denominator by the denominator.

Example:

Rationalise

5/√7

Multiply numerator and denominator by √7

5/√7 = (5/√7) x (√7/√7)

= 5√7/7

Rule 4:

\(a\sqrt{c}\pm b\sqrt{c}=(a\pm b)\sqrt{c}\)

Example:

To simplify,

5√6 + 4√6

5√6 + 4√6 = (5 + 4) √6

by the rule

= 9√6

Rule 5:

\(\frac{c}{a+b\sqrt{n}}\)

Multiply top and bottom by a-b √n

This rule enables us to rationalise the denominator.

Example:

To Rationalise

\(\frac{3}{2+\sqrt{2}}= \frac{3}{2+\sqrt{2}}\times \frac{2-\sqrt{2}}{2-\sqrt{2}}=\frac{6-3\sqrt{2}}{4-2}\) \(=\frac{6-3\sqrt{2}}{2}\)

Rule 6:

\(\frac{c}{a-b\sqrt{n}}\)

This rule enables you to rationalise the denominator.

Multiply top and bottom by a + b√n

Example:

To Rationalise

\(\frac{3}{2-\sqrt{2}}= \frac{3}{2-\sqrt{2}}\times \frac{2+\sqrt{2}}{2+\sqrt{2}}=\frac{6+3\sqrt{2}}{4-2}\) \(=\frac{6+3\sqrt{2}}{2}\)

Surds and Indices

Surds are the root values that cannot be written as whole numbers. Indices are the power or exponent of a value. For example, for 32, 2 is the index and 3 is the base. 

Suppose, we have a value √33. This value can be written as:

√3³ = (3³)^½ = 3^3/2 

Where 3/2 is the index.

Let us solve some examples here:

Example 1: 16^3/2 + 16^-3/2 = ?

We know, by laws of exponents,

a^m x aⁿ = a^m+n

a^-m = 1/a^m

⇒16^3/2 + 1/16^3/2

⇒(16^1/2)³ + 1/(16^1/2)³

⇒(4^{2 x 1/2})³ + 1/(4^{2 x 1/2})³

⇒4³ + 1/4³

⇒ 64 + 1/64

⇒ (64 x 64+ 1)/64

= (4096+1)/64

= 4097/64

Example 2: If (1/5 )^3a = 0.008 Find the value of ( 0.25)^a

Solution:

(1/5 )^3a = 0.008 = 8/1000 = 1/125 = (1/5 )³

Apply the Algebra Law and Solve the equation.

⇒ 3^a = 3

∴ a = 1

∴ ( 0.25 )^a = ( 0.25 )¹ = 0.25

How to Solve Surds?

You need to follow some rules to solve expressions that involve surds. One method is to rationalize the denominators, which helps to eject the surd in the denominator. Sometimes it may be mandatory to find the greatest perfect square factor to solve surds.

Problems and Solutions

Example 1: Write down the conjugate of 5√3 + √2

Solution:

The conjugate of  5√3 + √2 is 5√3 – √2.

Example 2: Rationalise the denominator: 1/[(8√11 )- (7√5)]

Solution:

Given:  1/[(8√11 )- (7√5)]

It is known that the conjugate of (8√11 )- (7√5) is (8√11 )+(7√5)

To rationalize the denominator of the given fraction, multiply the conjugate of denominator on both numerator and denominator.

=[1/[(8√11 )- (7√5)]]× [[(8√11 )+ (7√5)]/[(8√11 )+(7√5)]]

=[(8√11 )+ (7√5)]/[(8√11 )²-(7√5)²]

=[(8√11 )+ (7√5)]/[704- 245]

= [(8√11 )+ (7√5)]/459

Example 3: Multiply √7 x √2

Solution: √7 x √2 = √(7 x 2) = √14

Example 4: Divide √10 by √5.

Solution: √10/√5 = √(10/5) = √2

Example 5: Solve √x + 2√x.

Solution: √x + 2√x

= 3√x

Practice Questions

1. 2√2 x 3√5   [Answer: 6√10]
2. Write √28 in simplest form.
3. Solve (√10 + √3) (√10 – √3)
4. Solve by rationalising (5/(3 + √5).
5. Write in simplest form: 3/√5

Now to start practising problems and examples of surd based on rules mentioned above, please visit BYJU’S – The Learning App.