We are asked to find which expression results in a rational number. Let's analyze each option:

Option A: $\sqrt{2} \cdot \sqrt{18}$

Using the property of square roots, $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$: $\sqrt{2} \cdot \sqrt{18} = \sqrt{2 \cdot 18} = \sqrt{36} = 6$ Since 6 is a rational number, A is the correct answer.

Option B: $5 \cdot \sqrt{5}$

This expression results in $5\sqrt{5}$, which is irrational because $\sqrt{5}$ is irrational, and multiplying it by 5 does not make it rational.

Option C: $\sqrt{2} + \sqrt{2}$

This simplifies to $2\sqrt{2}$, which is irrational because $\sqrt{2}$ is irrational, and multiplying it by 2 does not make it rational.

Option D: $3\sqrt{2} + 2\sqrt{3}$

Both $\sqrt{2}$ and $\sqrt{3}$ are irrational, so adding these terms results in an irrational number.

Conclusion:

The only expression that results in a rational number is A: $\sqrt{2} \cdot \sqrt{18} = 6$.

Would you like further explanation or have any questions?

Related Questions:

1. What makes a number rational or irrational?
2. How do you multiply or add square roots?
3. How do properties of square roots simplify complex expressions?
4. Can multiplying an irrational number by a rational number ever result in a rational number?
5. What are some common irrational numbers?

Tip: Multiplying two square roots can simplify into a rational number if the product of the numbers inside the square roots is a perfect square.