Answer :
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]
### 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]
