How do you rationalize the denominator and simplify $(sqrtx-sqrty)/(sqrt(5x)+sqrt(6y))$ ?

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Answer 1 · 2018-01-21T17:07:00

See a solution process below:

To rationalize the denominator and simplify, first, multiply the fraction by the appropriate form of $1$ :

$(sqrt(5x) - sqrt(6y))/(sqrt(5x) - sqrt(6y)) xx (sqrt(x) - sqrt(y))/(sqrt(5x) + sqrt(6y)) =>$

$((sqrt(5x) - sqrt(6y))(sqrt(x) - sqrt(y)))/((sqrt(5x) - sqrt(6y))(sqrt(5x) + sqrt(6y))) =>$

$(sqrt(5x)sqrt(x) - sqrt(5x)sqrt(y) - sqrt(6y)sqrt(x) + sqrt(6y)sqrt(y))/((sqrt(5x))^2 + sqrt(5x)sqrt(6y) - sqrt(5x)sqrt(6y) - (sqrt(6y))^2) =>$

$(sqrt(5x * x) - sqrt(5x * y) - sqrt(6y * x) + sqrt(6y * y))/(5x + 0 - 6y) =>$

$(sqrt(5x^2) - sqrt(5xy) - sqrt(6xy) + sqrt(6y^2))/(5x - 6y) =>$

$(sqrt(5)x - sqrt(5xy) - sqrt(6xy) + sqrt(6)y)/(5x - 6y)$

If required, we can factor the numerator as:

$(sqrt(5)x - sqrt(5)sqrt(xy) - sqrt(6)sqrt(xy) + sqrt(6)y)/(5x - 6y) =>$

$(sqrt(5)(x - sqrt(xy)) - sqrt(6)(sqrt(xy) - y))/(5x - 6y)$

Or, we can also factor the numerator as:

$(sqrt(5)x - sqrt(5)sqrt(xy) - sqrt(6)sqrt(xy) + sqrt(6)y)/(5x - 6y) =>$

$(sqrt(5)x - sqrt(xy)(sqrt(5) + sqrt(6)) + sqrt(6)y)/(5x - 6y)$

How do you rationalize the denominator and simplify (sqrtx-sqrty)/(sqrt(5x)+sqrt(6y))(sqrtx-sqrty)/(sqrt(5x)+sqrt(6y))?