Let's analyze the numbers in the table to determine whether they are rational or irrational:

1. $-47.\overline{13}$: This represents a repeating decimal ($-47.131313...$). Repeating decimals are always rational numbers.

2. $-\frac{9}{2}$: This fraction simplifies to $-4.5$, which is a terminating decimal. Terminating decimals are rational numbers.

3. $-16\pi$: $\pi$ is an irrational number. Multiplying it by $-16$ still results in an irrational number.

4. $\sqrt{6}$: The square root of 6 is not a perfect square and cannot be simplified to a fraction, so it is irrational.

5. $-\sqrt{4}$: The square root of 4 is 2, and $-\sqrt{4}$ is $-2$, which is a rational number.

Now, I'll fill in the table based on this analysis:

• $-47.\overline{13}$: Rational
• $-\frac{9}{2}$: Rational
• $-16\pi$: Irrational
• $\sqrt{6}$: Irrational
• $-\sqrt{4}$: Rational

If you have any questions or need more details, feel free to ask! Here are some related questions to explore:

1. What are the properties of rational numbers?
2. How do you identify if a square root is rational or irrational?
3. Why are repeating decimals always considered rational?
4. Can a product of a rational and an irrational number ever be rational?
5. What is the difference between a terminating and a repeating decimal?

Tip: When dealing with square roots, remember that only the square roots of perfect squares are rational.