How do you rationalize the denominator and simplify (4-sqrt3)/(6+sqrty)?

1 Answer
Shwetank Mauria
Feb 12, 2017

(4-sqrt3)/(6+sqrty)=(24-4sqrty-6sqrt3+sqrt(3y))/(36-y)

Explanation:

To rationalize the denominator, we multiply numerator and denominator by the conjugate of its denominator .

Conjugate of an irrational number like sqrta+sqrtb is sqrta-sqrtb and that of p+sqrtq is p-sqrtq.

Here in (4-sqrt3)/(6+sqrty), denominator is 6+sqrty and assuming y is rational positive number, conjugate of denominator is 6-sqrty. As such

(4-sqrt3)/(6+sqrty)

= (4-sqrt3)/(6+sqrty)xx(6-sqrty)/(6-sqrty)

= ((4-sqrt3)(6-sqrty))/(6^2-(sqrty)^2)

= (4xx6-4sqrty-6sqrt3+sqrt(3y))/(6^2-(sqrty)^2)

= (24-4sqrty-6sqrt3+sqrt(3y))/(36-y)

Note - In case you have just sqrtp in denominator, just multiply numerator and denominator by sqrtp.

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