The problem asks us to evaluate the cube root of -729, i.e., find the value of $\sqrt[3]{-729}$. We need to determine if the cube root is a real number, and if so, find its value.

1. Problem Description

The problem asks us to evaluate the cube root of -729, i.e., find the value of $\sqrt[3]{-729}$. We need to determine if the cube root is a real number, and if so, find its value.

2. Solution Steps

We are looking for a number $x$ such that $x^3 = -729$.

We know that $9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$.

Therefore, $(-9)^3 = (-9) \times (-9) \times (-9) = 81 \times (-9) = -729$.

So, $\sqrt[3]{-729} = -9$.

3. Final Answer

The cube root of -729 is -

9. So, the answer is: $\sqrt[3]{-729} = -9$.