Question

what is the following quotient? \\(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\\) options: \\(\frac{\sqrt{30}+3\sqrt{2}+\sqrt{55}+\sqrt{33}}{8}\\), \\(\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{2}\\), \\(\frac{17}{8}\\), \\(-\frac{5}{2}\\)

Explanation:

Step1: Rationalize the Denominator

To simplify \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{5}-\sqrt{3}\).
\[ \begin{align*} \frac{(\sqrt{6}+\sqrt{11})(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}&=\frac{\sqrt{6}\times\sqrt{5}-\sqrt{6}\times\sqrt{3}+\sqrt{11}\times\sqrt{5}-\sqrt{11}\times\sqrt{3}}{(\sqrt{5})^2-(\sqrt{3})^2}\\ &=\frac{\sqrt{30}-\sqrt{18}+\sqrt{55}-\sqrt{33}}{5 - 3}\\ &=\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{2} \end{align*} \]
Wait, this doesn't match the given options. Maybe I made a mistake. Let's check the options again. Wait, maybe the original problem is \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}-\sqrt{3}}\)? Let's try that. Multiply numerator and denominator by \(\sqrt{5}+\sqrt{3}\):
\[ \begin{align*} \frac{(\sqrt{6}+\sqrt{11})(\sqrt{5}+\sqrt{3})}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})}&=\frac{\sqrt{6}\times\sqrt{5}+\sqrt{6}\times\sqrt{3}+\sqrt{11}\times\sqrt{5}+\sqrt{11}\times\sqrt{3}}{5 - 3}\\ &=\frac{\sqrt{30}+\sqrt{18}+\sqrt{55}+\sqrt{33}}{2}\\ &=\frac{\sqrt{30}+3\sqrt{2}+\sqrt{55}+\sqrt{33}}{2} \end{align*} \]
Still not matching. Wait, the options have denominators 8 and 2. Wait, maybe the original problem is \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\) and there was a miscalculation. Wait, let's check the options again. One of the options is \(\frac{\sqrt{30}+3\sqrt{2}+\sqrt{55}+\sqrt{33}}{8}\)? No, wait the first non - 17/8 option is \(\frac{\sqrt{30}+3\sqrt{2}+\sqrt{55}+\sqrt{33}}{8}\)? Wait, maybe I messed up the conjugate. Wait, no, let's recalculate the denominator: \((\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) = 5 - 3=2\). Wait, maybe the original problem is \(\frac{\sqrt{6}+\sqrt{11}}{2(\sqrt{5}+\sqrt{3})}\)? Let's multiply numerator and denominator by \(\sqrt{5}-\sqrt{3}\):
\[ \begin{align*} \frac{(\sqrt{6}+\sqrt{11})(\sqrt{5}-\sqrt{3})}{2\times((\sqrt{5})^2-(\sqrt{3})^2)}&=\frac{\sqrt{30}-\sqrt{18}+\sqrt{55}-\sqrt{33}}{2\times2}\\ &=\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{4} \end{align*} \]
No. Wait, the option \(\frac{\sqrt{30}+3\sqrt{2}+\sqrt{55}+\sqrt{33}}{8}\) – maybe the denominator was squared? Wait, no. Wait, maybe the original problem is \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\) and there's a miscalculation. Wait, let's check the value of \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\) numerically. \(\sqrt{6}\approx2.45\), \(\sqrt{11}\approx3.32\), numerator \(\approx2.45 + 3.32=5.77\). Denominator \(\sqrt{5}\approx2.24\), \(\sqrt{3}\approx1.73\), denominator \(\approx2.24+1.73 = 3.97\). So the value is approximately \(5.77\div3.97\approx1.45\). Now check the options: \(\frac{17}{8}=2.125\), \(\frac{\sqrt{30}+3\sqrt{2}+\sqrt{55}+\sqrt{33}}{8}\): \(\sqrt{30}\approx5.477\), \(3\sqrt{2}\approx4.242\), \(\sqrt{55}\approx7.416\), \(\sqrt{33}\approx5.744\). Sum: \(5.477 + 4.242+7.416 + 5.744\approx22.879\), divide by 8: \(\approx2.859\). \(\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{2}\): \(5.477-4.242 + 7.416-5.744\approx2.907\), divide by 2: \(\approx1.453\). Ah! This matches the approximate value of \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\approx1.45\). So there must be a sign error in the original problem's denominator. Maybe the denominator is \(\sqrt{5}-\sqrt{3}\) instead of \(\sqrt{5}+\sqrt{3}\). Let's redo the calculation with denominator \(\sqrt{5}-\sqrt{3}\):

Multiply numerator and denominator by \(\sqrt{5}+\sqrt{3}\):

\[ \begin{align*} \frac{(\sqrt{6}+\sqrt{11})(\sqrt{5}+\sqrt{3})}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})}&=\frac{\sqrt{30}+\sqrt{18}+\sqrt{55}+\sqrt{33}}{5 - 3}\\ &=\frac{\sqrt{30}+3\sqrt{2}+\sqrt{55}+\sqrt{33}}{2} \end{align*} \]
No, that's not. Wait, the option \(\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{2}\) gives approximately \((5.477-4.242 + 7.416-5.744)/2=(2.907)/2 = 1.453\), which is close to \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\approx(2.45 + 3.32)/(2.24 + 1.73)=5.77/3.97\approx1.45\). So the correct simplification (assuming a possible sign error in the denominator or numerator) leads to \(\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{2}\)

Step2: Verify with Approximations

As we saw above, the approximate value of \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\) is approximately \(1.45\), and the approximate value of \(\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{2}\) is also approximately \(1.45\), while the other options have different approximate values.

Answer:

\(\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{2}\) (the third option in the vertical list)