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To simplify \((-512)^{\frac{1}{3}}\), we need to find the cube root of \(-512\).

1. **Identify the cube root:** The cube root of a number \(a^{\frac{1}{3}}\) is a value that, when raised to the power of 3, results in \(a\).

2. **Calculate the cube root of -512:**
   \[
   (-512)^{\frac{1}{3}} = -8
   \]
   because \((-8) \times (-8) \times (-8) = -512\).

**Answer:** \((-512)^{\frac{1}{3}} = -8\).

Would you like further details or have any questions?

Here are five related questions you might be interested in:
1. What is the cube root of a positive number, like \(512^{\frac{1}{3}}\)?
2. How does taking an even root differ from taking an odd root of a negative number?
3. How would you calculate \((-512)^{\frac{2}{3}}\)?
4. What are some applications of cube roots in real-world problems?
5. How do you simplify other fractional exponents, like \(64^{\frac{1}{6}}\)?

**Tip:** When taking odd roots of negative numbers, the result is also negative, as shown here with \((-512)^{\frac{1}{3}} = -8\).

Learn how to simplify (-512)^(1/3) by finding the cube root of a negative number. This example covers key concepts in exponentiation and roots, focusing on cube roots. Suitable for students in Grades 8-10.

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