How do you simplify (3-4sqrt3)/(4sqrt5+3sqrt2)?

1 Answer
Shwetank Mauria
Jan 24, 2017

(3-4sqrt3)/(4sqrt5+3sqrt2)=(12sqrt5-9sqrt2-16sqrt15+12sqrt6)/62

Explanation:

To simplify (3-4sqrt3)/(4sqrt5+3sqrt2), what we need to do is to multiply numerator and denominator by the conjugate of its irrational denominator.

Conjugate of a irrational number (sqrta+-sqrtb) is (sqrta∓sqrtb). (If we just have sqrtp in denominator, then just multiplying numerator and denominator by sqrtp suffices.)

Hence, (3-4sqrt3)/(4sqrt5+3sqrt2)

= ((3-4sqrt3)(4sqrt5-3sqrt2))/((4sqrt5+3sqrt2)(4sqrt5-3sqrt2))

= (3(4sqrt5-3sqrt2)-4sqrt3(4sqrt5-3sqrt2))/((4sqrt5)^2-(3sqrt2)^2)

= (12sqrt5-9sqrt2-16sqrt15+12sqrt6)/(80-18)

= (12sqrt5-9sqrt2-16sqrt15+12sqrt6)/62

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