To simplify the expression [tex]\( 3 \sqrt{2} - \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{8}} \)[/tex], let's go through the steps meticulously:
### Step 1: Simplify each term separately
1. Simplify [tex]\( 3 \sqrt{2} \)[/tex]
This term is already in its simplest form.
2. Simplify [tex]\( -\frac{1}{\sqrt{2}} \)[/tex]
To rationalize the denominator, we multiply the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
-\frac{1}{\sqrt{2}} = -\frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = -\frac{\sqrt{2}}{2}
\][/tex]
3. Simplify [tex]\( -\frac{1}{\sqrt{8}} \)[/tex]
We start by simplifying [tex]\( \sqrt{8} \)[/tex]:
[tex]\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2}
\][/tex]
Now, rationalize the denominator:
[tex]\[
-\frac{1}{\sqrt{8}} = -\frac{1}{2 \sqrt{2}} = -\frac{1 \cdot \sqrt{2}}{2 \sqrt{2} \cdot \sqrt{2}} = -\frac{\sqrt{2}}{4}
\][/tex]
### Step 2: Combine all simplified terms
Rewrite the original expression using the simplified terms:
[tex]\[
3 \sqrt{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{4}
\][/tex]
### Step 3: Find a common denominator
To combine the terms, we convert them to have a common denominator (which is 4 in this case):
For [tex]\( 3 \sqrt{2} \)[/tex]:
[tex]\[
3 \sqrt{2} = \frac{3 \cdot 4 \sqrt{2}}{4} = \frac{12 \sqrt{2}}{4}
\][/tex]
For [tex]\( -\frac{\sqrt{2}}{2} \)[/tex]:
[tex]\[
-\frac{\sqrt{2}}{2} = -\frac{2 \sqrt{2}}{4}
\][/tex]
For [tex]\( -\frac{\sqrt{2}}{4} \)[/tex]:
[tex]\[
-\frac{\sqrt{2}}{4} = -\frac{\sqrt{2}}{4}
\][/tex]
### Step 4: Combine the terms over a common denominator
Now, we can combine:
[tex]\[
\frac{12 \sqrt{2}}{4} - \frac{2 \sqrt{2}}{4} - \frac{\sqrt{2}}{4}
\][/tex]
### Step 5: Simplify the fraction
Combine the numerators:
[tex]\[
\frac{12 \sqrt{2} - 2 \sqrt{2} - \sqrt{2}}{4} = \frac{(12 - 2 - 1) \sqrt{2}}{4} = \frac{9 \sqrt{2}}{4}
\][/tex]
Thus, the simplified form of the expression [tex]\( 3 \sqrt{2} - \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{8}} \)[/tex] is:
[tex]\[
\boxed{\frac{9 \sqrt{2}}{4}}
\][/tex]
