How to rationalize the denominator of the fraction (3*sqrt 2 - sqrt 6)/(4*sqrt 2 + sqrt 6)

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to rationalize the denominator of the fraction (3*sqrt 2 - sqrt 6)/(4*sqrt 2 + sqrt 6)

Use the relation (x - y)(x + y) = x^2 - y^2

(3*sqrt 2 - sqrt 6)/(4*sqrt 2 + sqrt 6)

=> (3*sqrt 2 - sqrt 6)(4*sqrt 2 - sqrt 6)/(4*sqrt 2 + sqrt 6)(4*sqrt 2 + sqrt 6)

=> (3*sqrt 2 - sqrt 6)(4*sqrt 2 - sqrt 6)/[(4*sqrt 2)^2 - (sqrt 6)^2]

=> (12*2 - 3*sqrt 12 - 4*sqrt 12 + 6)/[(4*sqrt 2)^2 - (sqrt 6)^2]

=> (12*2 - 3*sqrt 12 - 4*sqrt 12 + 6)/[16*2 - 6]

=> (30 - 14*sqrt 3)/ 26

=> (15 - 7*sqrt 3)/13

The required result is => (15 - 7*sqrt 3)/13

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To rationalize the denominator, we'll have to multiply both the numerator and denominator by the conjugate of the denominator:

(3sqrt2 - sqrt6)(4sqrt2 - sqrt6)/(4sqrt2 + sqrt6)(4sqrt2 - sqrt6)

The product of denominator returns a difference of two squares:

(3sqrt2 - sqrt6)(4sqrt2 - sqrt6)/(4^2*2 - 6)

We'll remove the brackets from the numerator:

(12*2 - 3sqrt12 - 4sqrt12 + 6)/(32-6)

We'll combine like terms inside brackets:

(30 - 7*2sqrt3)/26

(30-14sqrt3)/26 = 2(15 - 7sqrt3)/26

2(15 - 7sqrt3)/26 = (15 - 7sqrt3)/13

The rationalized fraction is (15 - 7sqrt3)/13.

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