The expression in the image is asking to simplify $(-512)^{\frac{1}{3}}$, which means the cube root of $-512$.

Solution:

1. Recall that $(-512)^{\frac{1}{3}}$ is asking for the number that, when raised to the third power, equals $-512$.
2. Since the base is negative and the exponent is a fraction with an odd denominator ($1/3$), the result will be negative. The cube root of a negative number is also negative.
3. The cube root of $512$ is $8$ because $8^3 = 8 \times 8 \times 8 = 512$.
4. Therefore, $(-512)^{\frac{1}{3}} = -8$.

Final Answer:

$\boxed{-8}$

Do you want further clarification or detailed steps for understanding cube roots?

Here are 5 related questions to expand your understanding:

1. What is the general rule for finding the cube root of a negative number?
2. How does the property of exponents apply to fractional exponents?
3. Can we compute $(-27)^{\frac{1}{3}}$ using the same logic?
4. What happens if the denominator of the exponent is even (e.g., $(-512)^{\frac{1}{4}}$)?
5. How can you simplify expressions like $x^{\frac{m}{n}}$ in general?

Tip: Remember, fractional exponents represent roots. For example, $a^{\frac{1}{n}}$ is the $n$-th root of $a$.