Rationalize the denominator:
y divided by (square root of 6 + square root of y)
To rationalize the denominator of the fraction y divided by the quantity square root of 6 plus square root of y, we need to eliminate any square roots from the denominator.
To do this, we'll use a technique called "rationalizing the denominator." The general idea is to multiply both the numerator and the denominator by a suitable expression that will eliminate the square root(s) in the denominator.
In this case, we have the denominator as the sum of two square roots, namely square root of 6 and square root of y. To eliminate these square roots, we'll use the conjugate of the denominator - which is just the same expression with the opposite sign in the middle. So, the conjugate of square root of 6 plus square root of y is square root of 6 minus square root of y.
To rationalize the denominator, we'll multiply both the numerator and the denominator of the fraction by the conjugate:
y divided by (square root of 6 + square root of y) * (square root of 6 - square root of y) / (square root of 6 - square root of y)
Now, we apply the distributive property to both the numerator and the denominator:
(y * (square root of 6 - square root of y)) / ((square root of 6)^2 - (square root of y)^2)
Simplifying further:
(y * (square root of 6 - square root of y)) / (6 - y)
Therefore, the rationalized form of the given expression is:
(y * (square root of 6 - square root of y)) / (6 - y)