Sure, let's simplify the given expressions step-by-step.
### 1.3.1. [tex]\(\frac{4 - \sqrt{2}}{2 + \sqrt{2}}\)[/tex]
To simplify this expression, we rationalize the denominator. We multiply both the numerator and the denominator by the conjugate of the denominator.
The conjugate of [tex]\( 2 + \sqrt{2} \)[/tex] is [tex]\( 2 - \sqrt{2} \)[/tex].
[tex]\[
\frac{4 - \sqrt{2}}{2 + \sqrt{2}} \times \frac{2 - \sqrt{2}}{2 - \sqrt{2}} = \frac{(4 - \sqrt{2})(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})}
\][/tex]
Now, we calculate the numerator and the denominator separately.
Numerator:
[tex]\[
(4 - \sqrt{2})(2 - \sqrt{2}) = 4 \cdot 2 + 4 \cdot (-\sqrt{2}) - \sqrt{2} \cdot 2 - \sqrt{2} \cdot (-\sqrt{2})
\][/tex]
[tex]\[
= 8 - 4\sqrt{2} - 2\sqrt{2} + 2 = 10 - 6\sqrt{2}
\][/tex]
Denominator:
[tex]\[
(2 + \sqrt{2})(2 - \sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2
\][/tex]
Putting the numerator and denominator together:
[tex]\[
\frac{10 - 6\sqrt{2}}{2} = 5 - 3\sqrt{2}
\][/tex]
So, [tex]\(\frac{4 - \sqrt{2}}{2 + \sqrt{2}} = 5 - 3\sqrt{2}.\)[/tex]
### 1.3.2. [tex]\(2 \sqrt{2} - 8 \sqrt{32}\)[/tex]
We can simplify the surd [tex]\(\sqrt{32}\)[/tex]:
[tex]\[
\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}
\][/tex]
Now we substitute this back into the original expression:
[tex]\[
2 \sqrt{2} - 8 \sqrt{32} = 2 \sqrt{2} - 8 \cdot 4\sqrt{2} = 2 \sqrt{2} - 32 \sqrt{2}
\][/tex]
[tex]\[
= (2 - 32) \sqrt{2} = -30 \sqrt{2}
\][/tex]
So, [tex]\(2 \sqrt{2} - 8 \sqrt{32} = -30 \sqrt{2}.\)[/tex]
### 1.3.3. [tex]\(2 \sqrt{27} + 6 \sqrt{48} + 6 \sqrt{75}\)[/tex]
We simplify each surd individually:
[tex]\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\][/tex]
[tex]\[
\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}
\][/tex]
[tex]\[
\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}
\][/tex]
Now substitute these back into the original expression:
[tex]\[
2 \sqrt{27} + 6 \sqrt{48} + 6 \sqrt{75} = 2 \cdot 3\sqrt{3} + 6 \cdot 4\sqrt{3} + 6 \cdot 5\sqrt{3}
\][/tex]
[tex]\[
= 6\sqrt{3} + 24\sqrt{3} + 30\sqrt{3}
\][/tex]
[tex]\[
= (6 + 24 + 30)\sqrt{3} = 60\sqrt{3}
\][/tex]
So, [tex]\(2 \sqrt{27} + 6 \sqrt{48} + 6 \sqrt{75} = 60 \sqrt{3}.\)[/tex]
Thus, the simplified expressions are:
1. [tex]\(\frac{4 - \sqrt{2}}{2 + \sqrt{2}} = 5 - 3\sqrt{2}\)[/tex]
2. [tex]\(2 \sqrt{2} - 8 \sqrt{32} = -30 \sqrt{2}\)[/tex]
3. [tex]\(2 \sqrt{27} + 6 \sqrt{48} + 6 \sqrt{75} = 60 \sqrt{3}\)[/tex]
