To solve the expression [tex]\(\frac{2}{\sqrt{2} + \sqrt{3} - \sqrt{5}}\)[/tex], let's work through the steps methodically.
1. Identify the expression:
[tex]\[
\frac{2}{\sqrt{2} + \sqrt{3} - \sqrt{5}}
\][/tex]
2. Rationalize the denominator:
To get rid of the radicals in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{2} + \sqrt{3} - \sqrt{5}\)[/tex] is [tex]\(\sqrt{2} + \sqrt{3} + \sqrt{5}\)[/tex].
3. Multiply both numerator and denominator by the conjugate:
[tex]\[
\frac{2}{\sqrt{2} + \sqrt{3} - \sqrt{5}} \cdot \frac{\sqrt{2} + \sqrt{3} + \sqrt{5}}{\sqrt{2} + \sqrt{3} + \sqrt{5}}
\][/tex]
This gives us:
[tex]\[
\frac{2(\sqrt{2} + \sqrt{3} + \sqrt{5})}{(\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} + \sqrt{3} + \sqrt{5})}
\][/tex]
4. Simplify the denominator:
We need to use the difference of squares formula to simplify:
[tex]\[
(\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} + \sqrt{3} + \sqrt{5})
\][/tex]
This simplifies to:
[tex]\[
(\sqrt{2} + \sqrt{3})^2 - (\sqrt{5})^2
\][/tex]
Calculate [tex]\((\sqrt{2} + \sqrt{3})^2\)[/tex]:
[tex]\[
(\sqrt{2} + \sqrt{3})^2 = (\sqrt{2})^2 + 2\sqrt{2}\sqrt{3} + (\sqrt{3})^2 = 2 + 6 + 3 = 5 + 2\sqrt{6}
\][/tex]
Therefore, the simplified form becomes:
[tex]\[
5 + 2\sqrt{6} - 5 = (2 + 3 - 5) = 0
\][/tex]
5. Combine the results:
The denominator simplifies to [tex]\(0\)[/tex], hence:
[tex]\[
\frac{2(\sqrt{2} + \sqrt{3} + \sqrt{5})}{0}
\][/tex]
The simplified result is undefined or tends towards infinity because we cannot divide by zero. In mathematical terms, it is represented by the symbol [tex]\( \text{zoo} \)[/tex] (complex infinity).
Thus:
The value of [tex]\(\frac{2}{\sqrt{2} + \sqrt{3} - \sqrt{5}}\)[/tex] is undefined or [tex]\( \text{zoo} \)[/tex].
