Sure, let's simplify the given expression step-by-step.
We need to simplify:
[tex]\[
\frac{3}{\sqrt{7} - \sqrt{2}} - \frac{3}{\sqrt{7} + \sqrt{2}}
\][/tex]
### Step 1: Rationalize the Denominators
For the first term:
[tex]\[
\frac{3}{\sqrt{7} - \sqrt{2}}
\][/tex]
Multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{7} + \sqrt{2}\)[/tex]:
[tex]\[
\frac{3}{\sqrt{7} - \sqrt{2}} \cdot \frac{\sqrt{7} + \sqrt{2}}{\sqrt{7} + \sqrt{2}} = \frac{3(\sqrt{7} + \sqrt{2})}{(\sqrt{7})^2 - (\sqrt{2})^2}
\][/tex]
Simplify the denominator:
[tex]\[
(\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5
\][/tex]
So, the first term rationalized is:
[tex]\[
\frac{3(\sqrt{7} + \sqrt{2})}{5} = \frac{3\sqrt{7} + 3\sqrt{2}}{5}
\][/tex]
For the second term:
[tex]\[
\frac{3}{\sqrt{7} + \sqrt{2}}
\][/tex]
Multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{7} - \sqrt{2}\)[/tex]:
[tex]\[
\frac{3}{\sqrt{7} + \sqrt{2}} \cdot \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}} = \frac{3(\sqrt{7} - \sqrt{2})}{(\sqrt{7})^2 - (\sqrt{2})^2}
\][/tex]
Simplify the denominator:
[tex]\[
(\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5
\][/tex]
So, the second term rationalized is:
[tex]\[
\frac{3(\sqrt{7} - \sqrt{2})}{5} = \frac{3\sqrt{7} - 3\sqrt{2}}{5}
\][/tex]
### Step 2: Simplify and Combine the Terms
Now we subtract the second term from the first term:
[tex]\[
\frac{3\sqrt{7} + 3\sqrt{2}}{5} - \frac{3\sqrt{7} - 3\sqrt{2}}{5}
\][/tex]
Combine the fractions:
[tex]\[
\frac{(3\sqrt{7} + 3\sqrt{2}) - (3\sqrt{7} - 3\sqrt{2})}{5} = \frac{3\sqrt{7} + 3\sqrt{2} - 3\sqrt{7} + 3\sqrt{2}}{5}
\][/tex]
Simplify the numerator:
[tex]\[
= \frac{3\sqrt{2} + 3\sqrt{2}}{5} = \frac{6\sqrt{2}}{5}
\][/tex]
So, the simplified result is:
[tex]\[
\frac{6\sqrt{2}}{5}
\][/tex]
