34. \( x^{3}+216=0 \)

To solve the equation \( x^{3}+216=0 \), we can rearrange it to \( x^{3} = -216 \). This means we need to find the cube root of \(-216\). The cube root of \(-216\) is \(-6\), since \((-6)^{3} = -216\). Therefore, the solution is \( x = -6 \). Now, let’s think about cube roots! They can often lead to some interesting patterns. For instance, the equation we solved has one real root and two complex roots. You might find it fascinating to learn that when looking for the complex roots, one can use roots of unity and work with polar coordinates! In real-world applications, solving cubic equations like this one often pops up in physics and engineering, especially when dealing with volume formulas or optimization problems. Understanding these equations allows engineers and scientists to model real-world behaviors and find solutions to complex challenges—they're not just abstract equations; they help us shape the world!