The problem asks to list the irrational numbers from the given set and select all that apply. Let's first go over what irrational numbers are. An irrational number is a number that cannot be expressed as a fraction of two integers (i.e., it cannot be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers). Irrational numbers have non-repeating, non-terminating decimal expansions.

Set provided:

$\left\{ -4\pi, -0.5333..., \frac{2}{13}, 0, 2, \sqrt{6}, \frac{43}{6}, 21.2, 88.777 \right\}$

Analyze each element:

1. $-4\pi$: This contains $\pi$, which is irrational. Any multiple of an irrational number is still irrational.

   • Irrational.

2. $-0.5333...$: This is a decimal that repeats, so it is a rational number.

   • Not irrational.

3. $\frac{2}{13}$: This is a fraction of two integers, meaning it's rational.

   • Not irrational.

4. $0$: Zero is a rational number.

   • Not irrational.

5. $2$: A whole number, so it is rational.

   • Not irrational.

6. $\sqrt{6}$: The square root of a non-perfect square is irrational.

   • Irrational.

7. $\frac{43}{6}$: A fraction of two integers, so it's rational.

   • Not irrational.

8. $21.2$: A terminating decimal, which is rational.

   • Not irrational.

9. $88.777$: This is a terminating decimal and thus is rational.

   • Not irrational.

Conclusion:

The irrational numbers in the set are $-4\pi$ and $\sqrt{6}$.

Answering the multiple choice:

• A: $\sqrt{6}$ – Irrational.
• D: $-4\pi$ – Irrational.

Correct answers: A, D.

Would you like further explanation or details on this?

Related Questions:

1. What makes a number irrational?
2. Why is $\pi$ considered irrational, and what other famous constants are irrational?
3. How can you identify irrational numbers from a set of decimals and fractions?
4. What is the difference between rational and irrational numbers?
5. Can an irrational number ever be expressed as a repeating decimal?

Tip:

Any number involving square roots of non-perfect squares or $\pi$ is usually irrational unless further simplified into a rational form.