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

Question

Accepted Answer

# 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:
$\frac{37 + 12\sqrt{5}}{59}$

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Question

Explanation:

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

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

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

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

Answer: