https://file.aiquickdraw.com/mathbot/file/17276632318763vlz48cb.jpg

Let's evaluate each expression.

### (a) \( (-256)^{\frac{1}{4}} \)
This is asking for the fourth root of \(-256\).

- The fourth root of a negative number is not a real number because no real number raised to an even power can result in a negative number.
- Therefore, the answer is: **Not a real number**.

### (b) \( (-1024)^{\frac{1}{5}} \)
This is asking for the fifth root of \(-1024\).

- The fifth root of a negative number does exist and is real because an odd root of a negative number is negative.
- \( 1024 = 2^{10} \), and since we want the fifth root, we compute:
  \[
  (-1024)^{\frac{1}{5}} = -(1024^{\frac{1}{5}}) = -2^2 = -4
  \]
- Therefore, the answer is: **-4**.

Do you want more details or have any questions? Here are some related questions:

1. What is the general rule for taking roots of negative numbers?
2. Can you take the square root of a negative number in the real number system?
3. How do you calculate fractional exponents?
4. What is the difference between even and odd roots of negative numbers?
5. What are imaginary numbers and how do they relate to this problem?

**Tip:** Odd roots of negative numbers are real, while even roots of negative numbers result in imaginary numbers.

Evaluate the following: (a) (-256)^(1/4), (b) (-1024)^(1/5)

Learn how to evaluate roots of negative numbers such as (-256)^(1/4) and (-1024)^(1/5). This problem involves understanding even and odd roots, and the distinction between real and imaginary numbers. Suitable for high school students in Grades 9-12.

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