Certainly! Let's simplify the expression [tex]\(\frac{2}{\sqrt{5} + \sqrt{3} + 2}\)[/tex].
To simplify the given fraction, we need to rationalize the denominator. This involves eliminating the square roots from the denominator, which can be accomplished by multiplying both the numerator and the denominator by the conjugate of the denominator.
The conjugate of [tex]\(\sqrt{5} + \sqrt{3} + 2\)[/tex] is [tex]\(\sqrt{5} + \sqrt{3} - 2\)[/tex].
Let’s proceed step by step:
1. Multiply the numerator and the denominator by the conjugate:
[tex]\[
\frac{2}{\sqrt{5} + \sqrt{3} + 2} \cdot \frac{\sqrt{5} + \sqrt{3} - 2}{\sqrt{5} + \sqrt{3} - 2}
\][/tex]
2. Apply the multiplication to both parts:
[tex]\[
\frac{2 (\sqrt{5} + \sqrt{3} - 2)}{(\sqrt{5} + \sqrt{3} + 2)(\sqrt{5} + \sqrt{3} - 2)}
\][/tex]
3. Expand the denominator using the difference of squares:
[tex]\[
(\sqrt{5} + \sqrt{3} + 2)(\sqrt{5} + \sqrt{3} - 2) = (\sqrt{5} + \sqrt{3})^2 - 2^2
\][/tex]
4. Calculate [tex]\((\sqrt{5} + \sqrt{3})^2\)[/tex] and [tex]\(2^2\)[/tex]:
[tex]\[
(\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
5. Combine both parts in the denominator:
[tex]\[
8 + 2\sqrt{15} - 4 = 4 + 2\sqrt{15}
\][/tex]
6. Simplify the denominator:
[tex]\[
4 + 2\sqrt{15}
\][/tex]
7. Now, substitute back into the fraction:
[tex]\[
\frac{2 (\sqrt{5} + \sqrt{3} - 2)}{4 + 2\sqrt{15}}
\][/tex]
8. Simplify by factoring out 2 from the numerator and 2 from the denominator:
[tex]\[
\frac{2 (\sqrt{5} + \sqrt{3} - 2)}{2(2 + \sqrt{15})} = \frac{\sqrt{5} + \sqrt{3} - 2}{2 + \sqrt{15}}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{2}{\sqrt{5} + \sqrt{3} + 2}\)[/tex] is:
[tex]\[
\frac{\sqrt{5} + \sqrt{3} - 2}{2 + \sqrt{15}}
\][/tex]
