ANSWER
[tex]\frac{5( \sqrt{11} + \sqrt{3} )} {8} [/tex]
EXPLANATION
The given rational function is
[tex] \frac{5}{ \sqrt{11} - \sqrt{3}} [/tex]
We need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of
[tex] \sqrt{11} - \sqrt{3} [/tex]
which is
[tex]\sqrt{11} + \sqrt{3} [/tex]
When we rationalize we obtain:
[tex] \frac{5( \sqrt{11} + \sqrt{3} )}{(\sqrt{11} - \sqrt{3} )( \sqrt{11} + \sqrt{3} )} [/tex]
The denominator is now a difference of two squares:
[tex](a - b)(a + b) = {a}^{2} - {b}^{2} [/tex]
We apply this property to get
[tex]\frac{5( \sqrt{11} + \sqrt{3} )}{( \sqrt{11}) ^{2} - ( \sqrt{3}) ^{2} )} [/tex]
[tex]\frac{5( \sqrt{11} + \sqrt{3} )}{11 - 3} [/tex]
This simplifies to
[tex]\frac{5( \sqrt{11} + \sqrt{3} )} {8} [/tex]
Or
[tex]\frac{5 } {8}\sqrt{11} + \frac{5 } {8}\sqrt{3} [/tex]
