OPERATIONS WITH RADICALS

Operations with radicals :

Addition, subtraction, multiplication and division of radical terms can be performed by laws of radicals. Let us see the rules one by one.

Multiplication of two radicals

Rule 1 :

Whenever we have two or more radical terms which are multiplied with same index, then we can put only one radical and multiply the terms inside the radical. 

Division of two radicals

Rule 2 :

Whenever we have two or more radical terms which are dividing with same index, then we can put only one radical and divide the terms inside the radical. 

Simplifying radicals with exponents

Rule 3 :

nth root of a can be written as a to the power 1/n. Whenever we have power to the power, we can multiply both powers.

Addition and subtraction of radicals

Addition and subtraction of two or more radical terms can be performed with like radicands only. Like radicand means a number which is inside root sign must be same but the number outside the radical may be different.

For example, 5√2 and 3√2 are like radical terms. Here the numbers inside the radicals are same.

How to move a square root to the other side

If the square root goes from one side of equal sign to the other side, it will become square

To simplify a number which is in radical sign we need to follow the steps given below.

Step 1:

Split the numbers in the radical sign as much as possible

Step 2:

If two same numbers are multiplying in the square root sign, we need to take only one number from the radical sign.

Step 3:

In case we have any number in front of radical sign already,we have to multiply the number taken out by the number in front of radical sign already.

Step 4:

If we have radical with the index n, (That is,  ) and the same term is multiplied by itself "n" times, then we need to take out only one term out from the radical.

For example, if we have radical with the index 3, (That is, ∛ ) and the same term is multiplied by itself three times, we need to take out only one term out from the radical.

Step 5:

Combining the like radical terms.

Let us see a example problem to understand this method.

Operations with radicals - Examples

Problem 1:

Simplify the following √5 x √18

Solution :

  =  √5 x √18

According to the laws of radical,

  =  √(5 x 18) ==> √(5 x 3 x 3)  ==> 3 √5

Problem 2 :

Simplify the following ∛7 x ∛8

Solution :

  =  ∛7 x ∛8

According to the laws of radical,

  = ∛(7 x 8) ==> ∛(7 x 2 x 2 x 2) ==> 2 ∛7 x 2 ==> 2 ∛14

Problem 3 :

Simplify the following 3√35 ÷ 2√7

Solution :

  =   3√35 ÷ 2√7

According to the laws of radical,

  =  (3/2) √(35/7) ==> (3/2)√5

Simplifying radical expressions

Problem 4 :

Simplify the following radical expression 

7 √30 + 2 √75 + 5 √50 

Solution :

                    = 7 √30 + 2 √75 + 5 √50 

First we have to split the given numbers inside the radical as much as possible.

                    =  √(5 x 2 x 3) + √(5 x 5 x 3) + √(5 x 5 x 2)

Here we have to keep √30 as it is.

                    =  √30 + 5 √3 + 5 √2 

Problem 5 :

Simplify the following radical expression 

√27 + √105 + √108 + √45

Solution :

                    = 3 √5 + 2√95 + 3√117 - √78

First we have to split the given numbers inside the radical as much as possible

                  =  √(3 x 3 x 3) + √(5 x 3 x 7) +

                       √(3 x 3 x 3 x 2 x 2) - √(5 x 5 x 3) 

                    =  3 √3 +  √105 + 3 x 2 √3 - 5 √3

                    =  3 √3 +  √105 + 6 √3 - 5 √3

                    = (3 + 6 - 5) √3 + √105

                    = 4 √3 + √105

Now let us see the next example of "Operations with radicals".

Problem 6 :

Simplify the following radical expression 

√45 + 3 √20 + √80 - 4 √40

Solution :

                    = √45 + 3 √20 + √80 - 4 √40

First we have to split the given numbers inside the radical as much as possible.

                     =  √(3 x 3 x 5) + √(2 x 2 x 5) +

                       √(5 x 2 x 2 x 2 x 2) - √(5 x 2 x 2 x 2) 

                    =  3 √5 + 2 √5 + 2 x 2 √5 - 2 √(2 x 5)

                    =  3 √5 + 2 √5 + 4 √5 - 2 √10

                    = (3 + 2 + 4) √5 - 2 √10

                    = 9 √5 - 2 √10

Now let us see the next example of "Operations with radicals".

Problem 7 :

Simplify the following radical expression 

3√5 + 2√95 + 3√117 - √78

Solution :

                    = 3 √5 + 2√95 + 3√117 - √78

First we have to split the given numbers inside the radical as much as possible

 =  3 √5 + 2 √(5 x 19) + 3 √(3 x 3 x 13) - √(3 x 2 x 13)

=  3 √5 + 2 √95 + 3 x 3 √13 - √78

=  3 √5 + 2 √95 + 9 √13 - √78

Now let us see the next example of "Operations with radicals".

Problem 8 :

Simplify the following radical expression

3 √32 - 2√8 + √50

Solution:

                    = 3 √32 - 2 √8 + √50

First we have to split the given numbers inside the radical as much as possible.

  =  3 √(2 x 2 x 2 x 2 x 2) - 2 √(2 x 2 x 2) + √(5 x 5 x 2) 

   =  (3 x 2 x 2 )√2 - (2 x 2) √2 + 5 √2 

  =  12 √2 - 4 √2 + 5 √2 

  = (12 + 5 - 4) √2

  = 13 √2

Now let us see the next example of "Operations with radicals".

Problem 9 :

Simplify the following radical expression

2 √12 - 3√27 - √243

Solution :

                    = 2 √12 - 3 √27 - √243 

First we have to split the given numbers inside the radical as much as possible.

  = 2 √(2 x 2 x 3) - 3 √(3 x 3 x 3) - √(3 x 3 x 3 x 3 x 3) 

   =  (2 x 2) √3 - (3 x 3) √3 - (3 x 3) √3 

  =  4 √3 - 9 √3 - 9 √3

  = ( 4 - 9 - 9 ) √3

  = -14 √3

Now let us see the next example of "Operations with radicals".

Problem 10 :

Simplify the following radical expression

√54 - √2500 - √24

Solution :

                    = √54 - √2500 - √24

First we have to split the given numbers inside the radical as much as possible.

= √(2 x 3 x 3 x 3)-√(5 x 5 x 5 x 5 x 2 x 2)-√(3 x 2 x 2 x 2)

  =  3 √(3 x 2) - (5 x 5 x 2) - (2 x 2) √(2 x 3)

  =  3 √6 - 50 - 4 √6

  =  (3 - 4) √6 - 50

  =  -√6 - 50

Now let us see the next example of "Operations with radicals".

Problem 11 :

Simplify the following radical expression

√45 - √25 - √80

Solution :

  =  √(5 x 3 x 3) - √(5 x 5) - √(5 x 2 x 2 x 2 x 2)

  =  3 √5 - 5 - 2 x 2√5

   =  3 √5 - 5 - 4√5

  =  -5 - 5

Problem 12 :

Simplify the following radical expression

5√95 - 2√50 - 3√180

Solution :

  =  5 √95 - 2 √50 - 3 √180

First we have to split the given numbers inside the radical as much as possible.

  =  5 √95  -  2 √(2 x 5 x 5) - 3 √(3 x 3 x 2 x 2 x 5) 

  =  5 √95 - (2 x 5) √2 - (3 x 2 x 3 )√5

  =  5 √95 - 10 √2 - 18 √5

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