How do you rationalize the denominator and simplify sqrt(3x)/(sqrtx-sqrt3)√3x√x−√3?

1 Answer
smendyka
Jul 5, 2017

See a solution process below:

Explanation:

To rationalize the denominator multiply the expression by this form of 11:

(sqrt(x) + sqrt(3))/(sqrt(x) + sqrt(3))√x+√3√x+√3

(sqrt(x) + sqrt(3))/(color(red)(sqrt(x)) + color(red)(sqrt(3))) xx sqrt(3x)/(color(blue)(sqrt(x)) - color(blue)(sqrt(3))) =>√x+√3√x+√3×√3x√x−√3⇒

(sqrt(3x)(sqrt(x) + sqrt(3)))/((color(red)(sqrt(x))color(blue)(sqrt(x))) - (color(red)(sqrt(x))color(blue)(sqrt(3))) + (color(red)(sqrt(3))color(blue)(sqrt(x))) - (color(red)(sqrt(3))color(blue)(sqrt(3)))) =>√3x(√x+√3)(√x√x)−(√x√3)+(√3√x)−(√3√3)⇒

(sqrt(3x)(sqrt(x) + sqrt(3)))/(x - (color(blue)(sqrt(3))color(red)(sqrt(x))) + (color(red)(sqrt(3))color(blue)(sqrt(x))) - 3) =>√3x(√x+√3)x−(√3√x)+(√3√x)−3⇒

(sqrt(3x)(sqrt(x) + sqrt(3)))/(x - 0 - 3) =>√3x(√x+√3)x−0−3⇒

(sqrt(3x)(sqrt(x) + sqrt(3)))/(x - 3) =>√3x(√x+√3)x−3⇒

Now, we can work on simplifying the numerator:

(color(red)(sqrt(3x))(sqrt(x) + sqrt(3)))/(x - 3) =>√3x(√x+√3)x−3⇒

((color(red)(sqrt(3x)) xx sqrt(x)) + (color(red)(sqrt(3x)) xx sqrt(3)))/(x - 3) =>(√3x×√x)+(√3x×√3)x−3⇒

(sqrt(3x^2) + sqrt(9x))/(x - 3) =>√3x2+√9xx−3⇒

(sqrt(3)x + 3sqrt(x))/(x - 3)√3x+3√xx−3

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