- Rationalize the denominator by multiplying the expression \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\) by the conjugate \(\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}\).
- Simplify the denominator using the difference of squares identity to obtain \(5 - 3 = 2\).
- Expand the numerator to get \(\sqrt{30} - \sqrt{18} + \sqrt{55} - \sqrt{33}\).
- Simplify \(\sqrt{18}\) to \(3\sqrt{2}\) to reach the final form \(\frac{\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}}{2}\), which is \(\boxed{\frac{\sqrt{30}-3 \sqrt{2}+\sqrt{55}-\sqrt{33}}{2}}\).
Explanation
- Analysis of the problem
The problem requires simplifying the quotient \(\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\). To simplify an expression with radicals in the denominator, a standard algebraic technique is to rationalize the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator, which allows us to eliminate the square roots from the denominator.
- Rationalizing the denominator
The denominator is \(\sqrt{5}+\sqrt{3}\). Its conjugate is \(\sqrt{5}-\sqrt{3}\). Multiplying the original quotient by this conjugate expression \(\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}\) (which is equal to 1) allows us to rewrite the expression without changing its value:
\[ \frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}} \times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}} \]
Using the difference of squares identity \((a+b)(a-b) = a^2 - b^2\) in the denominator gives us:
\[ (\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]
Now the denominator is a simple rational number.
- Simplifying the numerator and concluding the solution
Now we expand the numerator \(\left(\sqrt{6}+\sqrt{11}\right)\left(\sqrt{5}-\sqrt{3}\right)\) using the distributive property (FOIL method):
\[ \sqrt{6} \cdot \sqrt{5} - \sqrt{6} \cdot \sqrt{3} + \sqrt{11} \cdot \sqrt{5} - \sqrt{11} \cdot \sqrt{3} \]
This results in:
\[ \sqrt{30} - \sqrt{18} + \sqrt{55} - \sqrt{33} \]
We can simplify \(\sqrt{18}\) since \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\). Thus, the numerator becomes:
\[ \sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33} \]
Putting it all together, the entire expression is:
\[ \frac{\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}}{2} \]
Comparing this result to the given options, we identify the second choice as the correct one.
Examples
Rationalizing denominators is a fundamental skill in engineering and physics. For example, when calculating resistance in electrical circuits or modeling wave interference, formulas often result in complex radical expressions. Rationalizing these allows engineers to perform quick mental estimations or simplify complex equations to solve for variables efficiently without needing a calculator for every step. It turns an intimidating-looking expression into a manageable form that is easier to work with.