Question

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

Answer 1

`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)`