In this section, we will learn how to find the sum (or difference) of two or more surds.

Table of Contents

**How to add or subtract two or more surds**

The addition and subtraction of surds are the basic two operations on surds. The below steps need to be checked while adding two or more surds.

Step 1: First look into the sum (or difference) and check whether the surds involved in the sum (or difference) are in the simplest forms or not.

Step 2: If they are not in the simplest forms, then we have to convert them into their simplest forms.

Step 3: In this step, we will make two different lists of similar surds and dissimilar surds after making the simplest forms.

Step 4: Then we will add (or subtract) the rational coefficients of the similar surds and put the similar surd-factor beside this addition (or subtraction).

Step 5: The dissimilar surds will be connected with the desired $+$ or $-$ sign.

Step 6: The resultant surd will be the desired sum (or difference).

**Examples of addition of surds**

**Problem 1:** How to add two similar (or like) surds?

Solution: Lets consider two like surds $x\sqrt{a}$ and $y\sqrt{a}.$ Then their sum $=x\sqrt{a}+y\sqrt{a}$ $=(x+y)\sqrt{a}$

**Problem 2:** How to add two dissimilar (or unlike) surds?

Solution: Lets consider two unlike surds $x\sqrt{a}$ and $y\sqrt{b}$ where $a$ and $b$ are two different rational numbers. Then their sum is just $x\sqrt{a}+y\sqrt{b}.$ We cannot further simplify it.

**Examples of subtraction of surds**

**Problem 3:** How to subtract two like surds?

Solution: The subtraction of two similar surds $x\sqrt{a}$ and $y\sqrt{a}$ is $=x\sqrt{a}-y\sqrt{a}$ $(x-y)\sqrt{a}$

**Problem 4:** How to subtract two unlike surds?

Solution: The difference of two unlike surds $x\sqrt{a}$ and $y\sqrt{b}$ is $x\sqrt{a}-y\sqrt{b}.$ This cannot be further simplified.

Related Topics |

**Introduction to Surds****Order of Surds****Simple & Compound Surds****Pure & Mixed Surds****Like & Unlike Surds****Multiplication of Surds****Division of Surds****Conjugate Surds****Rationalisation of Surds**

## Question Answer on How to add Surds

**Question 1:** Find $\sqrt{18}+\sqrt{50}$

**Answer:**

At first, we will write both the square roots of 18 and 50 into their simplified radical forms. Then we will add the simplified forms. Note that we have

$\sqrt{18}=\sqrt{3 \times 3 \times 2}$ = 3√2.

$\sqrt{50}=\sqrt{5 \times 5 \times 2}$ = 5√2.

So we get that

√18 + √50 = 3√2 + 5√2 = (3+5)√2 = 8√2.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.