To find all real cube roots of \(-\frac{1}{216}\), we need to solve for \( x \) in the equation:

\[ x^3 = -\frac{1}{216} \]

1. **Identifying the Nature of the Roots:**
   - The cube root of a negative number is negative, as the cube of a negative number is negative.
   - For any real number \( x \), if \( x \) is negative, it has one real negative cube root.

2. **Calculating the Cube Root:**
   - The cube root of \(-\frac{1}{216}\) is \(-\sqrt[3]{\frac{1}{216}}\).

We can find the cube root of \(\frac{1}{216}\) first:

\[ \sqrt[3]{\frac{1}{216}} = \frac{\sqrt[3]{1}}{\sqrt[3]{216}} = \frac{1}{\sqrt[3]{216}} \]

Now, we find \(\sqrt[3]{216}\). Since \( 216 = 6^3 \):

\[ \sqrt[3]{216} = 6 \]

Therefore:

\[ \sqrt[3]{\frac{1}{216}} = \frac{1}{6} \]

So, the real cube root of \(-\frac{1}{216}\) is:

\[ -\frac{1}{6} \]

Thus, the only real cube root of \(-\frac{1}{216}\) is \(-\frac{1}{6}\).

Would you like any further details or have any other questions?

Here are some other questions you might be interested in:
1. How do you find the cube root of a positive number?
2. What are the properties of cube roots of negative numbers?
3. How can you express cube roots in exponential form?
4. Can you explain how to calculate cube roots using a calculator?
5. What are some real-life applications of finding cube roots?

**Tip:** When dealing with cube roots of fractions, it's often helpful to simplify the fraction first and then find the cube roots of the numerator and denominator separately.

Discover how to find all real cube roots of -1/216 with a detailed step-by-step solution. Learn about cube roots of negative numbers and their calculation method. Suitable for Grades 9-12 students and anyone interested in cube roots.

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