The expression you provided is \(\sqrt[4]{(-81)^3}\). Let's break it down step by step.

### Step 1: Evaluate \((-81)^3\)
\[
(-81)^3 = -81 \times -81 \times -81 = -531,441
\]

### Step 2: Take the fourth root of \(-531,441\)
The fourth root of a negative number is not a real number because any real number raised to an even power is positive. Therefore:

\[
\sqrt[4]{-531,441}
\]

This expression results in a complex number. The fourth root of \(-531,441\) can be expressed in terms of complex numbers. We would write it as:

\[
\sqrt[4]{-531,441} = 6\sqrt[4]{531,441} \times i = 27 \times \sqrt[4]{81} \times i = 9 \times 3 \times \sqrt{3} \times i = 27i\sqrt[4]{81}
\]

### Final Answer:
\[
\sqrt[4]{(-81)^3} = \sqrt[4]{-531,441} = 9\sqrt{3}i
\]

This represents a complex number where the imaginary unit \( i \) is included.

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Here are five related questions:
1. What is the cube of a negative number?
2. How do you calculate the fourth root of a number?
3. What is the significance of imaginary numbers in roots?
4. How can we represent roots of negative numbers?
5. What is the general method for solving roots with complex numbers?

**Tip:** When taking roots of negative numbers, remember that real roots exist only for odd roots. Even roots result in complex numbers.

Learn how to evaluate the fourth root of a complex number using the example of (-81)^3. This problem involves understanding exponentiation, roots of negative numbers, and complex numbers. Suitable for advanced high school or college-level mathematics students.

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