Addition and Subtraction of surds | How to add surds

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

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

Question Answer on How to add Surds

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


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.