**Properties of radicals worksheet :**

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

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

**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 :**

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 look at the the next problem on "Properties of radicals worksheet".

**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 look at the the next problem on "Properties of radicals worksheet".

**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 look at the the next problem on "Properties of radicals worksheet".

**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 look at the the next problem on "Properties of radicals worksheet".

**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 look at the the next problem on "Properties of radicals worksheet".

**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

**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

**Problem 13 :**

Solve for "x" :

2√x - 2 = 10

**Solution :**

2√x - 2 = 10

Add "2" on both sides

2√x = 12

Divide by "2" on both sides

√x = 6

x = 6²

x = 36

After having gone through the stuff given above, we hope that the students would have understood "Properties of radicals worksheet".

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