**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

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, **ⁿ√** ) 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.

**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

**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 "Simplify radical expressions mixed review".

**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 "Simplify radical expressions mixed review".

**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 "Simplify radical expressions mixed review".

**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 "Simplify radical expressions mixed review".

**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 "Simplify radical expressions mixed review".

**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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