To simplify the expression [tex]\(\frac{2}{\sqrt{5}+\sqrt{3}+2}\)[/tex], we aim to rationalize the denominator. This involves eliminating the square roots in the denominator. Here's a detailed step-by-step solution:
1. Express the Problem:
[tex]\[
\frac{2}{\sqrt{5} + \sqrt{3} + 2}
\][/tex]
2. Multiply by the Conjugate:
The goal in rationalizing the denominator is to clear out the square roots. However, the denominator has two irrational terms, [tex]\(\sqrt{5}\)[/tex] and [tex]\(\sqrt{3}\)[/tex], plus a rational term, [tex]\(2\)[/tex]. To rationalize such a complex denominator, we can use a method involving the sum and differences of squares.
Consider the conjugate of the whole denominator by creating a group with similar terms, say like [tex]\( (\sqrt{5} + \sqrt{3}) + 2\)[/tex]. We should multiply the numerator and the denominator by the expression containing both irrational parts as negative, which would be:
[tex]\[
\sqrt{5} + \sqrt{3} - 2
\][/tex]
3. Construct the Rationalizing Term and Multiply:
So, we multiply the numerator and the denominator by [tex]\((\sqrt{5} + \sqrt{3} - 2)\)[/tex]:
[tex]\[
\frac{2}{\sqrt{5} + \sqrt{3} + 2} \times \frac{\sqrt{5} + \sqrt{3} - 2}{\sqrt{5} + \sqrt{3} - 2}
\][/tex]
This gives us:
[tex]\[
\frac{2 (\sqrt{5} + \sqrt{3} - 2)}{(\sqrt{5} + \sqrt{3} + 2)(\sqrt{5} + \sqrt{3} - 2)}
\][/tex]
4. Simplify the Denominator (Use Difference of Squares):
To simplify the denominator, recall the formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]:
Let [tex]\(a = \sqrt{5} + \sqrt{3}\)[/tex] and [tex]\(b = 2\)[/tex], then:
[tex]\[
(\sqrt{5} + \sqrt{3})^2 - (2)^2
\][/tex]
Calculate the values:
[tex]\[
(\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}
\][/tex]
[tex]\[
(2)^2 = 4
\][/tex]
Substitute these into the expression:
[tex]\[
(\sqrt{5} + \sqrt{3} + 2)(\sqrt{5} + \sqrt{3} - 2) = (8 + 2\sqrt{15}) - 4 = 4 + 2\sqrt{15}
\][/tex]
5. Put the Rationalized Expression Together:
Having rationalised the denominator:
[tex]\[
\frac{2 (\sqrt{5} + \sqrt{3} - 2)}{4 + 2\sqrt{15}}
\][/tex]
6. Further Simplify the Expression:
To make simplification easier, factor the denominator:
[tex]\[
4 + 2\sqrt{15} = 2(2 + \sqrt{15})
\][/tex]
Therefore, the expression becomes:
[tex]\[
\frac{2 (\sqrt{5} + \sqrt{3} - 2)}{2(2 + \sqrt{15})} = \frac{\sqrt{5} + \sqrt{3} - 2}{2 + \sqrt{15}}
\][/tex]
Thus, the rationalized simplified form of the given expression is:
[tex]\[
\boxed{\frac{\sqrt{5} + \sqrt{3} - 2}{2 + \sqrt{15}}}
\][/tex]
