The question asks which expressions represent real numbers. Let's analyze each one:

1. $\sqrt{9}^{-\frac{1}{2}}$
2. $\sqrt{64}^{-\frac{1}{3}}$
3. $\sqrt{-9}$
4. $1 + \sqrt[3]{-8}$
5. $\sqrt[4]{-16}$
6. $i^4 = 4$

Analysis:

1. Expression 1: $\sqrt{9}^{-\frac{1}{2}}$

   • $\sqrt{9} = 3$, so this becomes $3^{-\frac{1}{2}} = \frac{1}{\sqrt{3}}$, which is a real number.

2. Expression 2: $\sqrt{64}^{-\frac{1}{3}}$

   • $\sqrt{64} = 8$, so this becomes $8^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{8}} = \frac{1}{2}$, which is a real number.

3. Expression 3: $\sqrt{-9}$

   • The square root of a negative number is imaginary, so this expression is not a real number (it's imaginary).

4. Expression 4: $1 + \sqrt[3]{-8}$

   • $\sqrt[3]{-8} = -2$, so this becomes $1 + (-2) = -1$, which is a real number.

5. Expression 5: $\sqrt[4]{-16}$

   • The fourth root of a negative number is imaginary when considering real numbers only, so this is not a real number.

6. Expression 6: $i^4 = 4$

   • $i^4 = 1$, not $4$, so this expression is incorrect in its given form. $i^4 = 1$ is a real number, but in this case, since it incorrectly equates to 4, the expression itself is problematic and not correct.

Final Answer:

The correct expressions that represent real numbers are:

1. $\sqrt{9}^{-\frac{1}{2}}$
2. $\sqrt{64}^{-\frac{1}{3}}$
3. $1 + \sqrt[3]{-8}$

Do you want further details or have any questions?

---

Here are some related questions for further exploration:

1. What is the difference between real and imaginary numbers?
2. How do you simplify expressions with fractional exponents?
3. What are the properties of cube roots for negative numbers?
4. Can a square root of a negative number ever be real?
5. What is the significance of $i$ in complex numbers?
6. How do higher roots (e.g., fourth roots) behave with negative numbers?
7. How can you verify if a number is real or imaginary?
8. Why does $i^4$ equal 1?

Tip: Always check the sign under the square root or other even roots to determine if the expression will result in a real or imaginary number.