In this section, we are going to learn how to simplify quotients with radicals.

Simplifying Quotients with Radicals - Examples

Example 1 :

Simplify the quotient :

6 / √5

Solution :

Multiply both numerator and denominator by √5 to get rid of the radical in the denominator.

6 / √5  =  (6/√5) ⋅ (√5/√5)

6 / √5  =  6√5 / 5

Example 2 :

Simplify the quotient :

2√3 / √6

Solution :

Simplify.

2√3/√6  =  2√3 / (√2 √3)

2√3/√6  =  2 / √2

On the right side, multiply both numerator and denominator by √2 to get rid of the radical in the denominator.

2√3/√6  =  (2/√2) ⋅ (√2/√2)

2√3/√6  =  2√2 / (√2 ⋅ √2)

2√3/√6  =  2√2 / 2

2√3/√6  =  √2

Example 3 :

Simplify the quotient :

√18 / (3√2)

Solution :

Simplify.

√18 / (3√2)  =  √(3 ⋅ 3 ⋅ 2) / (3√2)

√18 / (3√2)  =  3√2 / (3√2)

√18 / (3√2)  =  1

Example 4 :

Simplify the quotient :

√5 / √10

Solution :

Simplify.

√5 / √10  =  √5 / √(5 ⋅ 2)

√5 / √10  =  √5 / (√5 ⋅ 2)

√5 / √10  =  1 / √2

On the right side, multiply both numerator and denominator by √2 to get rid of the radical in the denominator.

√5 / √10  =  (1/√2) ⋅ (√2/√2)

√5 / √10  =  √2 / 2

Example 5 :

Simplify the quotient :

/ √72

Solution :

Decompose 72 into prime factor using synthetic division. √72  =  √(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3)

√72  =  2 ⋅ 3 ⋅ √2

√72  =  6√2

Then, we have

9 / √72  =  9 / 6√2

Simplify.

9 / √72  =  3 / 2√2

On the right side, multiply both numerator and denominator by √2 to get rid of the radical in the denominator.

9 / √72  =  (3/2√2) ⋅ (√2/√2)

9 / √72  =  3√2 / (2√2 ⋅ √2)

9 / √72  =  3√2 / 4 After having gone through the stuff given above, we hope that the students would have understood, "Simplifying Quotients with Radicals".

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