Let's analyze and solve this problem step-by-step:

We are given a number line with points labeled $C, A, E, D, B$, and we need to match the points to the expressions provided in the options. Each expression involves square roots, so we will evaluate each one and place it on the number line to see if it corresponds with the given positions.

Approximate Values:

1. $\sqrt{2} \approx 1.414$
2. $\sqrt{3} \approx 1.732$
3. $\sqrt{8} \approx 2.828$
4. $\sqrt{10} \approx 3.162$
5. $\sqrt{12} \approx 3.464$

Evaluate each expression:

• (a) $A = -\sqrt{8}$:

  • $-\sqrt{8} \approx -2.828$, which is close to -3. This matches the position of $A$ on the number line (near -3).

• (b) $B = 3\sqrt{2}$:

  • $3\sqrt{2} = 3 \times 1.414 \approx 4.242$, which is slightly above 4. This does not match the position of $B$, which is at 6 on the number line.

• (c) $C = -\sqrt{10}$:

  • $-\sqrt{10} \approx -3.162$, which is close to -3. This matches the position of $C$, which is also near -3.

• (d) $D = \sqrt{12}$:

  • $\sqrt{12} \approx 3.464$, which is close to 3.5. This corresponds well with the position of $D$, which is marked at 4 on the number line.

• (e) $E = -\sqrt{3}$:

  • $-\sqrt{3} \approx -1.732$, which is between -1 and -2. This does not match the position of $E$, which is positive (near 1) on the number line.

Conclusion:

The accurately placed values are:

• (a) $A = -\sqrt{8}$
• (c) $C = -\sqrt{10}$
• (d) $D = \sqrt{12}$

Would you like a deeper explanation of any part of this solution?

Here are 5 related questions to expand your knowledge:

1. How can we approximate square roots to more decimal places?
2. What are the properties of irrational numbers like square roots?
3. How can we visually represent irrational numbers on a number line?
4. How can we simplify square roots, like $\sqrt{8}$ into $2\sqrt{2}$?
5. What is the geometric significance of square roots in coordinate systems?

Tip: Always break down square root problems into smaller approximations when evaluating them on a number line.