To divide radical expressions, we have to take separate roots for both numerator and denominator.

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.

Case 1 :

If we have prime number in the denominator, we have to multiply both numerator and denominator by the same prime number along with radical sign.

Case 2 :

If the denominator is in any one of the following forms,  ± √b, ± √b, a√b ± c, we have to rationalize the denominator. For that we have to multiply the given fraction by its conjugate.

Let us look into some examples problems based on the above concepts.

Example 1 :

Simplify the radical expression given below

√15/5 √10

Solution :

=  √15/5 √10

=  (1/5) √15/√10

=  (1/5) √(15/10)

Dividing 15 and 10 by 5, we get 3/2

=  (1/5) √(3/2)

=  3/5√2

Example 2 :

Simplify the radical expression given below

√8/√100

Solution :

=  √8/√100

=  √(2 ⋅ 2 ⋅ 2)/√(5 ⋅ 5 ⋅ 2 ⋅ 2)

=  2√2/(5 ⋅ 2)

=  2√2/10

=  √2/5

Example 3 :

Simplify the radical expression given below

√6/√27

Solution :

=  √6/√27

=  √(6/27)

=  √2/9

=  √2/(3 ⋅ 3 ⋅ 3)

=  √2/3√3

Let us look into the next example problem on "Divide radical expressions".

Example 4 :

Simplify the radical expression given below

√(3x2y3)/4√(5xy3)

Solution :

=  √(3x2y3)/4√(5xy3)

=  (1/4)√(3x2y3/5xy3)

=  (1/4)√(3x/5)

=  (1/4) √3x/5

Example 5 :

Simplify the radical expression given below

3/(4 + √5)

Solution :

=  3/(4 + √5)

In order to simplify the given radical expression, we need to multiply both numerator and denominator by the conjugate of 4 + √5

Conjugate of 4 + √5 is 4 - √5

=  [3/(4 + √5)] x [(4 - √5) / (4 - √5)]

=  [3(4 - √5)/(4 + √5) (4 - √5)]

=  [3(4 - √5)/(42 - √52)]

=  [3(4 - √5)/(16 - 5)]

=  (12 - 3√5)/11

=  (12/11) - (3√5/11)

Example 6 :

Simplify the radical expression given below

5/(5 + 3√3)

Solution :

=  5/(5 + 3√3)

In order to simplify the given radical expression, we need to multiply both numerator and denominator by the conjugate of 5 + 3√3

Conjugate of 5 + 3√3 is 5 - 3√3

=  [5/(5 + 3√3)] x [(5 - 3√3) / (5 - 3√3)]

=  [5(5 - 3√3)/(5 + 3√3(5 - 3√3)]

=  [5(5 - 3√3)/(52 - (3√3)2)]

=  [5(5 - 3√3)/(25 - (9(3))]

=  [5(5 - 3√3)/(25 - 27)]

=  [5(5 - 3√3)/(-2)]

=  [(25 - 15√3)/(-2)]

=  15√3 - 25/2

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