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[tex]\displaystyle\\\frac{5}{\sqrt{11}-\sqrt{3}}=?\\\\\\\text{We rationalize the denominator.}\\\\\frac{5}{\sqrt{11}-\sqrt{3}}=\frac{5(\sqrt{11}+\sqrt{3})}{(\sqrt{11}-\sqrt{3})(\sqrt{11}+\sqrt{3})}=\frac{5(\sqrt{11}+\sqrt{3})}{(11-3)}=\boxed{\bf\frac{5\sqrt{11}+5\sqrt{3})}{8}}[/tex]

Answer:

The correct option is b) [tex]\frac{5\sqrt{11}+5\sqrt{3}}{8}[/tex]

Step-by-step explanation:

We need to find the quotient of [tex]\frac{5}{\sqrt{11}-\sqrt{3}}[/tex],

Rationalizing the above,

By multiply and divide by conjugate of its denominator,

[tex]\frac{5}{\sqrt{11}-\sqrt{3}} \times \frac{\sqrt{11}+\sqrt{3}}{\sqrt{11}+\sqrt{3}}[/tex]

[tex]\frac{5(\sqrt{11}+\sqrt{3})}{(\sqrt{11}-\sqrt{3})(\sqrt{11}+\sqrt{3})}[/tex]

Since, [tex](a+b)(a-b)=a^{2}-b^{2}[/tex]

[tex]\frac{5\sqrt{11}+5\sqrt{3}}{(11-3)}[/tex]

simplify,

[tex]\frac{5\sqrt{11}+5\sqrt{3}}{8}[/tex]

Therefore, the correct option is b) [tex]\frac{5\sqrt{11}+5\sqrt{3}}{8}[/tex]

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