You can use the rationalization method in which we multiply the fraction with conjugate of the denominator.
The quotient of the given fraction is given as
![\dfrac{\sqrt{30} + \sqrt{55} - 3\sqrt{2} - \sqrt{33}}{2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Csqrt%7B30%7D%20%2B%20%5Csqrt%7B55%7D%20-%203%5Csqrt%7B2%7D%20-%20%5Csqrt%7B33%7D%7D%7B2%7D)
<h3>How to rationalize a fraction?</h3>
Suppose the given fraction is ![\dfrac{a}{b + c}](https://tex.z-dn.net/?f=%5Cdfrac%7Ba%7D%7Bb%20%2B%20c%7D)
Then the conjugate of the denominator is given by b - c
Thus, rationalizing the fraction will give us
![\dfrac{a}{b+c} = \dfrac{a}{b+c} \times \dfrac{b-c}{b-c} = \dfrac{a(b-c)}{b^2 - c^2}\\\\\\(since \: \: (x+y)(x-y) = x^2 - y^2 )](https://tex.z-dn.net/?f=%5Cdfrac%7Ba%7D%7Bb%2Bc%7D%20%3D%20%5Cdfrac%7Ba%7D%7Bb%2Bc%7D%20%5Ctimes%20%5Cdfrac%7Bb-c%7D%7Bb-c%7D%20%3D%20%5Cdfrac%7Ba%28b-c%29%7D%7Bb%5E2%20-%20c%5E2%7D%5C%5C%5C%5C%5C%5C%28since%20%5C%3A%20%5C%3A%20%28x%2By%29%28x-y%29%20%3D%20x%5E2%20-%20y%5E2%20%29)
We actually rationalize just for the use of that later denominator or numerator(if they seem to be helpful).
Remember that
thus, multiplying it with the fraction doesn't change its value, and just change the way how it looks. We assume that b-c is not 0
<h3>Using above method for getting the quotient of the given fraction</h3>
![\dfrac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}} = \dfrac{\sqrt{6} + \sqrt{11}}{\sqrt{5} + \sqrt{3}} \times \dfrac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \dfrac{(\sqrt{6} + \sqrt{11}) \times ( \sqrt{5} - \sqrt{3})}{(\sqrt{5})^2 - (\sqrt{3})^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Csqrt%7B6%7D%20%20%2B%20%5Csqrt%7B11%7D%7D%7B%5Csqrt%7B5%7D%20%2B%20%5Csqrt%7B3%7D%7D%20%3D%20%5Cdfrac%7B%5Csqrt%7B6%7D%20%20%2B%20%5Csqrt%7B11%7D%7D%7B%5Csqrt%7B5%7D%20%2B%20%5Csqrt%7B3%7D%7D%20%20%5Ctimes%20%5Cdfrac%7B%5Csqrt%7B5%7D%20%20-%20%5Csqrt%7B3%7D%7D%7B%5Csqrt%7B5%7D%20-%20%5Csqrt%7B3%7D%7D%20%20%3D%20%20%5Cdfrac%7B%28%5Csqrt%7B6%7D%20%20%2B%20%5Csqrt%7B11%7D%29%20%5Ctimes%20%28%20%5Csqrt%7B5%7D%20%20-%20%5Csqrt%7B3%7D%29%7D%7B%28%5Csqrt%7B5%7D%29%5E2%20%20-%20%28%5Csqrt%7B3%7D%29%5E2%7D)
Simplifying the fraction:
![\dfrac{(\sqrt{6} + \sqrt{11}) \times ( \sqrt{5} - \sqrt{3})}{(\sqrt{5})^2 - (\sqrt{3})^2} = \dfrac{\sqrt{30} + \sqrt{55} - 3\sqrt{2} - \sqrt{33}}{2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%28%5Csqrt%7B6%7D%20%20%2B%20%5Csqrt%7B11%7D%29%20%5Ctimes%20%28%20%5Csqrt%7B5%7D%20%20-%20%5Csqrt%7B3%7D%29%7D%7B%28%5Csqrt%7B5%7D%29%5E2%20%20-%20%28%5Csqrt%7B3%7D%29%5E2%7D%20%3D%20%5Cdfrac%7B%5Csqrt%7B30%7D%20%2B%20%5Csqrt%7B55%7D%20-%203%5Csqrt%7B2%7D%20-%20%5Csqrt%7B33%7D%7D%7B2%7D)
Thus,
The quotient of the given fraction is given as
![\dfrac{\sqrt{30} + \sqrt{55} - 3\sqrt{2} - \sqrt{33}}{2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Csqrt%7B30%7D%20%2B%20%5Csqrt%7B55%7D%20-%203%5Csqrt%7B2%7D%20-%20%5Csqrt%7B33%7D%7D%7B2%7D)
Learn more about rationalizing fractions here:
brainly.com/question/14261303