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Adding and Subtracting Radicals

An expression such as $\sqrt{2}$, $\sqrt[3]{4}$, $\sqrt{a+b}$ that exhibits a radical sign, is referred to as a "radical". The word Radical is derived from the Latin word "radix" which means "root". The radical symbol ($\sqrt{}$), is used to indicate the square root or $n^{th}$ root.The number inside the radical are called as Radicand.
For example:
Radicand
The condition under which radicals can be added or subtracted are much the same as the condition for letters in an algebraic expression. The like radicals can be add or subtract. If the radicals are unlike radicals, try to simplify one or both of the radicals.
For example: $\sqrt{2} + \sqrt{5}$ cannot be added.
$\sqrt{3} + \sqrt[3]{3}$ cannot be added.

Definition: A radical is the principal $r^{th}$ root of k where r is a positive integer greater than one and k ia a real number.
The operation of the radicals are opposite to the exponents.
For example: $\sqrt[r]{k}$ in this case r is called as index, √ is called as radical sign, and k is called as radicand.

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How to Add and Subtract Radicals

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Below are the steps for simplifying the add and subtract radicals.
1) The sum or difference of the reduced radicals can be obtained by adding or subtracting their co-efficient.
2) Remember only like radicals can be add or subtract.
3) If the radicals are unlike radicals, try to simplify one or both of the radicals.
4) Once the radicals are simplified, add or subtract the numbers directly preceding the radicals that are same.

Solved Examples

Question 1: Simplify 4 $\sqrt{5}$ + 6$\sqrt{5}$ - 2$\sqrt{5}$
Solution:
 
Given 4 $\sqrt{5}$ + 6 $\sqrt{5}$ - 2 $\sqrt{5}$
= 10 $\sqrt{5}$ - 2 $\sqrt{5}$
= 8 $\sqrt{5}$
 

Question 2: 2 $\sqrt{3}$ + 6$\sqrt{3}$ - 3 $\sqrt{3}$
Solution:
 
Given 2 $\sqrt{3}$ + 6$\sqrt{3}$ - 3 $\sqrt{3}$
= 8 $\sqrt{3}$ - 3 $\sqrt{3}$
= 5 $\sqrt{3}$
 

Question 3: 12 $\sqrt{5}$ + 7 $\sqrt{5}$ + 2 $\sqrt{5}$
Solution:
 
Given 12 $\sqrt{5}$ + 7 $\sqrt{5}$ + 2 $\sqrt{5}$
= 19 $\sqrt{5}$ + 2 $\sqrt{5}$
= 21$\sqrt{5}$
 

Question 4: 2 $\sqrt{75}$ + 4$\sqrt{3}$ - 8$\sqrt{27}$
Solution:
 
Given 2 $\sqrt{75}$ + 4 $\sqrt{3}$ - 8 $\sqrt{27}$
= 2 $\sqrt{25 \times 3}$ + 4 $\sqrt{3}$ - 8 $\sqrt{9 \times 3}$
= 2 $\times$ 5 $\sqrt{3}$ + 4 $\sqrt{3}$ - 8 $\times$ 3 $\sqrt{3}$
=10 $\sqrt{3}$ + 4 $\sqrt{3}$ - 24 $\sqrt{3}$
= 14 $\sqrt{3}$ - 24 $\sqrt{3}$
= -10 $\sqrt{3}$
 

Adding and Subtracting Radical Expressions

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Two or more radical expressions are called as like radicals if they have the same index and the same radicand.
Two radical expressions that are like radicals can be added or subtracted by adding or subtracting their coefficients.

Solved Examples

Question 1: 17 $\sqrt{7}$ + 4 $\sqrt{7}$ - 12 $\sqrt{7}$
Solution:
 
Given 17 $\sqrt{7}$ + 4 $\sqrt{7}$ - 12 $\sqrt{7}$
                   = 21 $\sqrt{7}$ - 12 $\sqrt{7}$
                   = 9 $\sqrt{5}$
 

Question 2: 8 $\sqrt{32}$ - 3 $\sqrt{18}$ + 2 $\sqrt{3}$
Solution:
 
Given equation is
8 $\sqrt{32}$ - 3 $\sqrt{18}$ + 2 $\sqrt{3}$
= 8 $\sqrt{16 \times 2}$ - 3 $\sqrt{9 \times 2}$ + 2 $\sqrt{3}$
=8 $\times$ 4$\sqrt{2}$ - 3 $\times$ 3 $\sqrt{2}$ +2 $\sqrt{3}$
= 32 $\sqrt{2}$ - 9 $\sqrt{2}$ + 2 $\sqrt{3}$
= 23 $\sqrt{2}$ + 2 $\sqrt{3}$
 

Question 3: 2 $\sqrt{25}$ - 3 $\sqrt{80}$ + 3 $\sqrt{36}$
Solution:
 
Given 2 $\sqrt{25}$ - 3 $\sqrt{80}$ + 3$\sqrt{36}$
= 2 $\times$ 5 - 3 $\times$ 4 $\sqrt{5}$ + 3 $\times$ 6
= 10 - 12 $\sqrt{5}$ + 18
= 28 - 12 $\sqrt{5}$
= 4 [ 7 - 3 $\sqrt{5}$ ]
 

Question 4: 4x $\sqrt{25}$ - 3x $\sqrt{80}$ + 2x $\sqrt{36}$
Solution:
 
Given 4x $\sqrt{25}$ - 3x $\sqrt{80}$ + 2x $\sqrt{36}$
= 4x $\times$ 5 - 3x $\times$ 4 $\sqrt{5}$ + 12x
= 4x - 12x $\sqrt{5}$ + 12x
= 16x - 12x $\sqrt{5}$
= 4x [ 4 - 3 $\sqrt{5}$ ]
 

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