How do you rationalize the denominator and simplify sqrtx/(sqrtx+sqrt3)?

1 Answer
Alan P.
Jun 30, 2016

sqrt(x)/(sqrt(x)+sqrt(3))=color(blue)((x-sqrt(3x))/(x-3))

Explanation:

For the general expression: (a+b)
the conjugate is (a-b)
and the product of an expression times its conjugate is
color(white)("XXX")(a+b)*(a-b)=a^2-b^2

If a and/or b are square roots this lets us get rid of the square roots.

For the given example sqrt(x)/(sqrt(x)+sqrt(3))

multiplying both the numerator and denominator by the conjugate of (sqrt(x)+sqrt(3)) lets us remove the square roots from the denominator.

color(white)("XXX")sqrt(x)/(sqrt(x)+sqrt(3))xx(sqrt(x)-sqrt(3))/(sqrt(x)-sqrt(3))

color(white)("XXX")=(sqrt(x)*(sqrt(x)-sqrt(3)))/(x-3)

color(white)("XXX")=(x-sqrt(3x))/(x-3)

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