Certainly! Let's explore the given mathematical expressions and simplify each step by step.
### Expression 1
[tex]\[
\frac{2 - \sqrt{8}}{4 + \sqrt{12}}
\][/tex]
First, we simplify the square roots in the expression:
[tex]\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}
\][/tex]
[tex]\[
\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}
\][/tex]
Now, substitute these back into the original expression:
[tex]\[
\frac{2 - 2\sqrt{2}}{4 + 2\sqrt{3}}
\][/tex]
The simplified form of this expression is:
[tex]\[
\frac{2 - 2\sqrt{2}}{2\sqrt{3} + 4}
\][/tex]
### Expression 2
[tex]\[
\frac{\sqrt{3} - \sqrt{6}}{4}
\][/tex]
This expression does not require further simplification. It is already in its simplest form.
### Expression 3
[tex]\[
\frac{2 + \sqrt{3} - 2\sqrt{2} - \sqrt{6}}{4}
\][/tex]
This expression is already quite simplified, but for the sake of clarity, we recognize that no further factorization or simplification is possible within the radicals involved:
[tex]\[
\frac{2 + \sqrt{3} - 2\sqrt{2} - \sqrt{6}}{4}
\][/tex]
### Expression 4
[tex]\[
2 - \sqrt{3} - 2\sqrt{2} + \sqrt{6}
\][/tex]
This expression is also in its simplest form as there are no further factors or common terms to simplify.
### Expression 5
[tex]\[
\frac{-2 - \sqrt{3} + 2\sqrt{2} + \sqrt{6}}{2}
\][/tex]
For this expression, we can simplify the numerator, but there is not much we can factor out or combine further:
[tex]\[
\frac{-2 - \sqrt{3} + 2\sqrt{2} + \sqrt{6}}{2}
\][/tex]
In conclusion, the simplified forms of the given expressions are as follows:
1. [tex]\(\frac{2 - 2\sqrt{2}}{2\sqrt{3} + 4}\)[/tex]
2. [tex]\(\frac{\sqrt{3} - \sqrt{6}}{4}\)[/tex]
3. [tex]\(\frac{2 + \sqrt{3} - 2\sqrt{2} - \sqrt{6}}{4}\)[/tex]
4. [tex]\(2 - \sqrt{3} - 2\sqrt{2} + \sqrt{6}\)[/tex]
5. [tex]\(\frac{-2 - \sqrt{3} + 2\sqrt{2} + \sqrt{6}}{2}\)[/tex]
