To simplify the expression [tex]\(\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} + \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\)[/tex], we must evaluate each term separately before adding them together. Here’s a step-by-step simplification process:
1. Simplify the first term [tex]\(\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(\sqrt{3} + \sqrt{2}\)[/tex]:
[tex]\[
\frac{(\sqrt{3}+\sqrt{2})(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}
\][/tex]
- The denominator simplifies using the difference of squares:
[tex]\[
(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1
\][/tex]
- The numerator becomes:
[tex]\[
(\sqrt{3} + \sqrt{2})^2 = (\sqrt{3})^2 + 2\sqrt{3}\sqrt{2} + (\sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6}
\][/tex]
- So, the first term is:
[tex]\[
\frac{5 + 2\sqrt{6}}{1} = 5 + 2\sqrt{6}
\][/tex]
2. Simplify the second term [tex]\(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\)[/tex]:
- Multiply the numerator and the denominator by the conjugate of the denominator, [tex]\(\sqrt{3} - \sqrt{2}\)[/tex]:
[tex]\[
\frac{(\sqrt{3}-\sqrt{2})(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}
\][/tex]
- The denominator simplifies using the difference of squares:
[tex]\[
(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1
\][/tex]
- The numerator becomes:
[tex]\[
(\sqrt{3} - \sqrt{2})^2 = (\sqrt{3})^2 - 2\sqrt{3}\sqrt{2} + (\sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6}
\][/tex]
- So, the second term is:
[tex]\[
\frac{5 - 2\sqrt{6}}{1} = 5 - 2\sqrt{6}
\][/tex]
3. Add the simplified terms:
- The final expression is:
[tex]\[
(5 + 2\sqrt{6}) + (5 - 2\sqrt{6})
\][/tex]
- Simplifying this, we notice the [tex]\(\pm 2\sqrt{6}\)[/tex] terms cancel out:
[tex]\[
5 + 5 = 10
\][/tex]
Thus, the simplified expression for [tex]\(\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} + \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\)[/tex] is:
[tex]\[
10
\][/tex]
