For all x>0 and y>0, the radical expression (fraction: let me try to spell it out) square root of X on top of fraction, denominator is 3 Square root of X minus the square root of Y is equivalent to:
sqrt x/(3 sqrtx - sqrt y)
= sqrtx(3 sqrtx + sqrty)/(9x - y)
= [3x + sqrt(xy)]/(9x -y)
To simplify the expression (sqrt(x))/(3sqrt(x) - sqrt(y)) where x > 0 and y > 0, we can rationalize the denominator.
Rationalizing the denominator involves getting rid of any square roots in the denominator.
To do this, we multiply the numerator and denominator of the fraction by the conjugate of the denominator, which is 3sqrt(x) + sqrt(y).
Let's multiply the numerator and denominator by 3sqrt(x) + sqrt(y) now:
(sqrt(x)) * (3sqrt(x) + sqrt(y))
-----------------------------------
(3sqrt(x) - sqrt(y)) * (3sqrt(x) + sqrt(y))
Expanding the denominator using the difference of squares formula (a^2 - b^2 = (a + b)(a - b)):
(sqrt(x)) * (3sqrt(x) + sqrt(y))
-----------------------------------
(3sqrt(x))^2 - (sqrt(y))^2
Simplifying the denominator:
(sqrt(x)) * (3sqrt(x) + sqrt(y))
-----------------------------------
9 * x - y
Therefore, the simplified expression is:
(sqrt(x))(3sqrt(x) + sqrt(y))
-----------------------------
9x - y