Question

rationalize the denominator.
\\(\frac{4\sqrt{2} + \sqrt{10}}{8\sqrt{2} - \sqrt{10}}\\)

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of \(8\sqrt{2} - \sqrt{10}\) is \(8\sqrt{2} + \sqrt{10}\). So we multiply the fraction by \(\frac{8\sqrt{2} + \sqrt{10}}{8\sqrt{2} + \sqrt{10}}\):
\[
\frac{(4\sqrt{2} + \sqrt{10})(8\sqrt{2} + \sqrt{10})}{(8\sqrt{2} - \sqrt{10})(8\sqrt{2} + \sqrt{10})}
\]

Step2: Expand the numerator using the distributive property (FOIL method).

\[

$$\begin{align*} &(4\sqrt{2})(8\sqrt{2}) + (4\sqrt{2})(\sqrt{10}) + (\sqrt{10})(8\sqrt{2}) + (\sqrt{10})(\sqrt{10})\\ =& 32\times2 + 4\sqrt{20} + 8\sqrt{20} + 10\\ =& 64 + 12\sqrt{20} + 10\\ =& 74 + 12\sqrt{4\times5}\\ =& 74 + 24\sqrt{5} \end{align*}$$

\]

Step3: Expand the denominator using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\).

\[

$$\begin{align*} &(8\sqrt{2})^2 - (\sqrt{10})^2\\ =& 64\times2 - 10\\ =& 128 - 10\\ =& 118 \end{align*}$$

\]

Step4: Simplify the fraction by dividing numerator and denominator by their greatest common divisor (2 in this case).

\[
\frac{74 + 24\sqrt{5}}{118}=\frac{37 + 12\sqrt{5}}{59}
\]

Answer:

\(\frac{37 + 12\sqrt{5}}{59}\)