# PROPERTIES OF SQUARE ROOTS WORKSHEET

Properties of square roots worksheet :

Worksheet on properties of square roots is much useful to the students who would like to practice problems on radicals.

## Properties of square roots worksheet - Problems

1)  Simplify the following √5 x √18

2)  Simplify the following ∛7 x ∛8

3)  Simplify the following 3√35 ÷ 2√7

4)  Simplify the following radical expression

7 √30 + 2 √75 + 5 √50

5)  Simplify the following radical expression

√27 + √105 + √108 + √45

6)  Simplify the following radical expression

√45 + 3 √20 + √80 - 4 √40

7)  Simplify the following radical expression

3√5 + 2√95 + 3√117 - √78

8)  Simplify the following radical expression

3 √32 - 2√8 + √50

9)  Simplify the following radical expression

2 √12 - 3√27 - √243

10)  Simplify the following radical expression

√54 - √2500 - √24

11)  Simplify the following radical expression

√45 - √25 - √80

12)  Simplify the following radical expression

5√95 - 2√50 - 3√180

13)  Solve for "x" :

2√x - 2  =  10

## Properties of square roots worksheet - Solution

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

Now let us look at the the next problem on  "Properties of square roots worksheet".

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 look at the the next problem on  "Properties of square roots worksheet".

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 look at the the next problem on  "Properties of square roots worksheet".

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 look at the the next problem on  "Properties of square roots worksheet".

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 look at the the next problem on  "Properties of square roots worksheet".

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

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

Problem 13 :

Solve for "x" :

2√x - 2  =  10

Solution :

2√x - 2  =  10

2√x  =  12

Divide by "2" on both sides

√x  =  6

x  =  6²

x  =  36

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