# SIMPLIFY RADICAL EXPRESSIONS MIXED REVIEW

Simplify radical expressions mixed review :

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

We can add or subtract only like radicals. Like radicals means we have same number inside the radical sign, the coefficient may be different but the number inside the square root and the order of the square root must be same.

2 √2 + 5 √2 = 7 √2

√2 + √7 = can not be simplified

## Multiplication and division of two radicals :

We can multiply or divide two radicals which are having same order only. The number inside the radical may be different but the order of the radical must be same.

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:

Let us see a example problem to understand this method.

## Simplify radical expressions mixed review - 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

Problem 4 :

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 :

√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

Problem 6 :

√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 "Simplify radical expressions mixed review".

Problem 7 :

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 "Simplify radical expressions mixed review".

Problem 8 :

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 "Simplify radical expressions mixed review".

Problem 9 :

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 "Simplify radical expressions mixed review".

Problem 10 :

√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 "Simplify radical expressions mixed review".

Problem 11 :

√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 :

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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