what is the following quotient?\n$\frac{5}{sqrt{11}-sqrt{3}}$\n$\frac{5sqrt{11}-5sqrt{3}}{8}$\n$\frac{5sqrt{1…

what is the following quotient?\n$\frac{5}{sqrt{11}-sqrt{3}}$\n$\frac{5sqrt{11}-5sqrt{3}}{8}$\n$\frac{5sqrt{11}+5sqrt{3}}{8}$\n$\frac{5}{8}$\n$\frac{5sqrt{2}}{4}$

what is the following quotient?\n$\frac{5}{sqrt{11}-sqrt{3}}$\n$\frac{5sqrt{11}-5sqrt{3}}{8}$\n$\frac{5sqrt{11}+5sqrt{3}}{8}$\n$\frac{5}{8}$\n$\frac{5sqrt{2}}{4}$

Answer

Explanation:

Step1: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{11}+\sqrt{3}$: $\frac{5(\sqrt{11}+\sqrt{3})}{(\sqrt{11}-\sqrt{3})(\sqrt{11}+\sqrt{3})}$

Step2: Expand the denominator

Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, we have $(\sqrt{11}-\sqrt{3})(\sqrt{11}+\sqrt{3})=(\sqrt{11})^{2}-(\sqrt{3})^{2}=11 - 3=8$

Step3: Expand the numerator

$5(\sqrt{11}+\sqrt{3}) = 5\sqrt{11}+5\sqrt{3}$ So the fraction becomes $\frac{5\sqrt{11}+5\sqrt{3}}{8}$

Answer:

$\frac{5\sqrt{11}+5\sqrt{3}}{8}$