To simplify the expression [tex]\(\frac{2}{\sqrt{7}+\sqrt{2}} + \frac{1}{\sqrt{7}-\sqrt{2}}\)[/tex] and write it as a single fraction, we will follow these steps:
1. Rationalize the denominators:
Rationalizing the denominator means multiplying the numerator and denominator of each fraction by the conjugate of the denominator.
2. Simplify each fraction:
After rationalizing, simplify each fraction separately.
3. Combine the fractions:
Express both fractions with a common denominator and combine them.
Let's begin by rationalizing the denominators:
### Step 1: Rationalize the denominators
#### For [tex]\(\frac{2}{\sqrt{7} + \sqrt{2}}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{7} - \sqrt{2}\)[/tex]:
[tex]\[
\frac{2}{\sqrt{7}+\sqrt{2}} \cdot \frac{\sqrt{7}-\sqrt{2}}{\sqrt{7}-\sqrt{2}} = \frac{2(\sqrt{7}-\sqrt{2})}{(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})}
\][/tex]
The denominator becomes:
[tex]\[
(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2}) = 7 - 2 = 5
\][/tex]
So, the fraction simplifies to:
[tex]\[
\frac{2(\sqrt{7} - \sqrt{2})}{5} = \frac{2\sqrt{7} - 2\sqrt{2}}{5}
\][/tex]
#### For [tex]\(\frac{1}{\sqrt{7} - \sqrt{2}}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{7} + \sqrt{2}\)[/tex]:
[tex]\[
\frac{1}{\sqrt{7} - \sqrt{2}} \cdot \frac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}} = \frac{\sqrt{7}+\sqrt{2}}{(\sqrt{7}-\sqrt{2})(\sqrt{7}+\sqrt{2})}
\][/tex]
The denominator becomes:
[tex]\[
(\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2}) = 7 - 2 = 5
\][/tex]
So, the fraction simplifies to:
[tex]\[
\frac{\sqrt{7} + \sqrt{2}}{5}
\][/tex]
### Step 2: Combine the fractions
Now we add the two simplified fractions:
[tex]\[
\frac{2\sqrt{7} - 2\sqrt{2}}{5} + \frac{\sqrt{7} + \sqrt{2}}{5}
\][/tex]
Combine them over a common denominator:
[tex]\[
\frac{(2\sqrt{7} - 2\sqrt{2}) + (\sqrt{7} + \sqrt{2})}{5}
\][/tex]
Combine the like terms in the numerator:
[tex]\[
\frac{2\sqrt{7} + \sqrt{7} - 2\sqrt{2} + \sqrt{2}}{5} = \frac{3\sqrt{7} - \sqrt{2}}{5}
\][/tex]
### Step 3: Conclusion
The expression [tex]\(\frac{2}{\sqrt{7}+\sqrt{2}} + \frac{1}{\sqrt{7}-\sqrt{2}}\)[/tex] as a single fraction is:
[tex]\[
\boxed{\frac{3\sqrt{7} - \sqrt{2}}{5}}
\][/tex]
