To solve the expression [tex]\(\frac{2}{3+\sqrt{2}+\sqrt{5}}\)[/tex], we need to rationalize the denominator. This involves eliminating the irrational numbers (square roots) from the denominator.
To achieve this, we will multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a sum of square roots like [tex]\(3 + \sqrt{2} + \sqrt{5}\)[/tex] is usually taken by changing the signs of the square roots, so in this case, it will be [tex]\(3 - \sqrt{2} - \sqrt{5}\)[/tex].
Here are the steps in detail:
1. Identify the original expression:
[tex]\[
\frac{2}{3+\sqrt{2}+\sqrt{5}}
\][/tex]
2. Multiply both the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{2}{3+\sqrt{2}+\sqrt{5}} \times \frac{3-\sqrt{2}-\sqrt{5}}{3-\sqrt{2}-\sqrt{5}} = \frac{2(3-\sqrt{2}-\sqrt{5})}{(3+\sqrt{2}+\sqrt{5})(3-\sqrt{2}-\sqrt{5})}
\][/tex]
3. Expand the numerator:
[tex]\[
2(3 - \sqrt{2} - \sqrt{5}) = 6 - 2\sqrt{2} - 2\sqrt{5}
\][/tex]
4. Expand the denominator using the distributive property (a-b)(a+b) to handle the square roots:
[tex]\[
(3+\sqrt{2}+\sqrt{5})(3-\sqrt{2}-\sqrt{5})
\][/tex]
Let’s break it down using the distributive property:
[tex]\[
(3+\sqrt{2}+\sqrt{5})(3-\sqrt{2}-\sqrt{5})
\][/tex]
= [tex]\(3(3) + 3(-\sqrt{2}) + 3(-\sqrt{5}) + \sqrt{2}(3) + \sqrt{2}(-\sqrt{2}) + \sqrt{2}(-\sqrt{5}) + \sqrt{5}(3) + \sqrt{5}(-\sqrt{2}) + \sqrt{5}(-\sqrt{5})\)[/tex]
Group like terms together:
[tex]\[
= 3(3) + 3(-\sqrt{2}) + 3(-\sqrt{5}) + 3\sqrt{2} + \sqrt{2}(-\sqrt{2}) + \sqrt{2}(-\sqrt{5}) + 3\sqrt{5} + \sqrt{5}(-\sqrt{2}) + \sqrt{5}(-\sqrt{5})
\][/tex]
Simplify terms involving square roots:
[tex]\[
= 9 - 3\sqrt{2} - 3\sqrt{5} + 3\sqrt{2} - 2 -\sqrt{10} + 3\sqrt{5} - \sqrt{10} - 5
\][/tex]
Combine and cancel out terms:
[tex]\[
= 9 - 2 - 5 - 2\sqrt{10}
= 2 - 2\sqrt{10}
\][/tex]
5. Combine the simplified numerator and denominator:
[tex]\[
\frac{6 - 2\sqrt{2} - 2\sqrt{5}}{2 - 2\sqrt{10}}
= \frac{2(3 - \sqrt{2} - \sqrt{5})}{2(1 - \sqrt{10})}
= \frac{3 - \sqrt{2} - \sqrt{5}}{1 - \sqrt{10}}
\][/tex]
So, the final result is:
[tex]\(\frac{3 - \sqrt{2} - \sqrt{5}}{1 - \sqrt{10}}\)[/tex]
