https://file.aiquickdraw.com/mathbot/file/172547637303364nzdlus.jpg

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.

Based on the image, determine whether each of the given numbers is rational or irrational.

Analyze the given numbers, including repeating decimals, fractions, square roots, and multiples of pi, to determine whether each is rational or irrational. Suitable for students in Grades 8-10.

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