To simplify the given expression $3 \times \frac{\sqrt{3}}{2} + 2 \times \frac{1}{\sqrt{2}}$, let's proceed step by step.

Expression:

$3 \times \frac{\sqrt{3}}{2} + 2 \times \frac{1}{\sqrt{2}}$

1. Simplify each part:

   • First part: $3 \times \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}$
   • Second part: $2 \times \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}}$

2. Rationalize the second term: To simplify $\frac{2}{\sqrt{2}}$, multiply both the numerator and denominator by $\sqrt{2}$: $\frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}$

3. Final expression: Now the expression becomes: $\frac{3\sqrt{3}}{2} + \sqrt{2}$

This is the simplified form of the expression. If needed, this can be further approximated numerically.

Would you like to see the numerical approximation or have any other questions about the process?

Related Questions:

1. How do you rationalize denominators with square roots?
2. How can you approximate $\sqrt{2}$ and $\sqrt{3}$ numerically?
3. What are some common methods for simplifying square roots in algebraic expressions?
4. How does multiplying by a square root help in simplifications?
5. How does this example relate to trigonometric simplifications?

Tip: Rationalizing a denominator eliminates radicals in the denominator, making expressions easier to handle in many cases.