Laws of Surds

A surd is an irrational number that cannot be simplified into a rational number. It is written in root form, as it has no exact rational value. If a is a positive rational number and n is a positive integer such that n√a is irrational, then na is called a surd. In this guide, we will learn the definition of surds, the laws of surds, and related examples to build a strong understanding of the concept.

Table of Contents

• What are the Laws of Surds?
• Solved Examples on the Laws of Surds

What are the Laws of Surds

In this section, you will learn about the laws of surds and how they help in simplifying irrational numbers in root form.

Note : 

• Unlike surds cannot be added or subtracted:   a+b≠(a+b),a−b≠(a−b).

• Surds can be written in exponential form. an=a1n.
  
  

Solved Examples on  the Laws of Surds

Example 1: Simplify 3√12 + 2√27 − √48.
Solution: 3√12  = 3 × 2√3 = 6√3
2√27 = 2 × 3√3 = 6√3
√48 = 4√3
3√12 + 2√27 − √48 = 6√3 + 6√3 - 4√3 = 8√3

Example 2: Multiply 4√3  and 2√5
Solution: 4√3 × 2√5 = 4 × 2 × √3 × √5
= 8 × √(3 × 5)
= 8√15

Example 3: Rationalise the denominator of 1/(3 + √2).
Solution: Multiply the numerator and denominator by (3-√2)

  [13+2=13+2×3−23−2=3−2(3+2)(3−2)=3−27∴13+2=3−27