$$ 3\sqrt2 $$-$$  $$$$ \frac{1}{\sqrt2} $$

The expression you've provided is:

\[
3\sqrt{2} - \frac{1}{\sqrt{2}}
\]

To simplify this expression, we can rewrite the second term:

\[
\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\]

Now the expression becomes:

\[
3\sqrt{2} - \frac{\sqrt{2}}{2}
\]

We can factor out $\sqrt{2}$ from both terms:

\[
\sqrt{2}\left(3 - \frac{1}{2}\right)
\]

Simplifying the expression inside the parentheses:

\[
\sqrt{2}\left(\frac{6}{2} - \frac{1}{2}\right) = \sqrt{2} \cdot \frac{5}{2} = \frac{5\sqrt{2}}{2}
\]

So the simplified expression is:

\[
\frac{5\sqrt{2}}{2}
\]

Would you like more details, or do you have any questions about this? Here are some related questions you might find interesting:

1. How would this expression change if the first term was $4\sqrt{3}$ instead of $3\sqrt{2}$?
2. What is the decimal approximation of the simplified expression $\frac{5\sqrt{2}}{2}$?
3. How would you rationalize the denominator in the expression $\frac{1}{\sqrt{2}}$ from the start?
4. What happens to the expression if the square root were in the denominator instead?
5. How would you graph the expression $y = 3\sqrt{2} - \frac{1}{\sqrt{2}}$ as a constant function?

**Tip:** When working with square roots, it's often helpful to rationalize the denominator to simplify the expression.

Learn how to simplify the expression $3\sqrt{2} - \frac{1}{\sqrt{2}}$ with detailed steps. Understand algebraic techniques such as factoring and rationalizing denominators. Suitable for high school students.

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