To solve this problem, we need to compare the two numbers given: $-\sqrt[3]{22}$ and $-\pi$, and then identify which number line representation is correct.

Step-by-Step Breakdown:

1. Approximate the values of the numbers:

   • $\pi$ is approximately $3.1416$, so $-\pi \approx -3.1416$.
   • $\sqrt[3]{22}$ (the cube root of 22) is approximately $2.802$, so $-\sqrt[3]{22} \approx -2.802$.

2. Comparison of the two values:

   • $-\pi$ is roughly $-3.1416$, which is smaller (more negative) than $-\sqrt[3]{22}$, which is about $-2.802$.
   • Therefore, on a number line, $-\pi$ will be to the left of $-\sqrt[3]{22}$.

Analyze the number lines:

Now, we need to find the diagram where $-\pi$ is to the left of $-\sqrt[3]{22}$ on the number line.

• The second number line in the image, which is highlighted, places $-\pi$ to the left of $-\sqrt[3]{22}$. This is the correct choice.

Answer:

The second number line is the correct representation because $-\pi$ is less than $-\sqrt[3]{22}$.

Would you like further details, or do you have any questions?

Here are 5 related questions to expand on the concept:

1. How do you compute cube roots of other numbers?
2. What are some ways to estimate the value of $\pi$ more accurately?
3. What would the positions of $\sqrt{22}$ and $\pi$ look like on the positive side of a number line?
4. How can you solve $\sqrt[3]{x}$ for other values of $x$?
5. Why is $\pi$ an irrational number, and how does that affect its position on the number line?

Tip: When comparing irrational numbers like $\pi$ with other numbers, approximating them with a few decimal places can help visualize their relative positions on the number line.