Let's break this question into parts:

Part (a) Plot $\frac{3}{\sqrt{2}}$ and $3 - \sqrt{2}$

To plot these values on a number line, we need to calculate their approximate decimal values:

1. Calculate $\frac{3}{\sqrt{2}}$: $\sqrt{2} \approx 1.414$ $\frac{3}{\sqrt{2}} \approx \frac{3}{1.414} \approx 2.121$

2. Calculate $3 - \sqrt{2}$: $3 - \sqrt{2} \approx 3 - 1.414 \approx 1.586$

So, on the number line:

• $\frac{3}{\sqrt{2}} \approx 2.1$
• $3 - \sqrt{2} \approx 1.6$

Part (b) Compare $\frac{3}{\sqrt{2}}$ and $3 - \sqrt{2}$:

From the above calculations: $\frac{3}{\sqrt{2}} \approx 2.1 \quad \text{and} \quad 3 - \sqrt{2} \approx 1.6$ Clearly: $\frac{3}{\sqrt{2}} > 3 - \sqrt{2}$

Final Answer:

• Plot the points at approximately 1.6 and 2.1 on the number line.
• For the comparison, $\frac{3}{\sqrt{2}} > 3 - \sqrt{2}$.

Would you like more detailed calculations or additional examples?

Here are 5 related questions to explore:

1. How do you simplify expressions like $\frac{a}{\sqrt{b}}$?
2. What are the steps to rationalize the denominator of a fraction involving square roots?
3. How is the square root approximated using a calculator?
4. Can you prove why $\sqrt{2}$ is an irrational number?
5. How do square root values affect real-world measurements?

Tip: Always rationalize the denominator to keep mathematical expressions in their simplest form.