The problem asks to simplify the expression $\sqrt[3]{-27x^{18}}$.

1. Problem Description

The problem asks to simplify the expression $\sqrt[3]{-27x^{18}}$.

2. Solution Steps

We need to simplify the cube root of $-27x^{18}$.

First, we can express $-27$ as $(-3)^3$. So we have:

$\sqrt[3]{-27x^{18}} = \sqrt[3]{(-3)^3(x^{18})}$

Since $\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$, we can rewrite this as:

$\sqrt[3]{(-3)^3}\sqrt[3]{x^{18}}$

The cube root of $(-3)^3$ is $-3$:

$\sqrt[3]{(-3)^3} = -3$

To find the cube root of $x^{18}$, we can use the property $\sqrt[n]{x^m} = x^{\frac{m}{n}}$:

$\sqrt[3]{x^{18}} = x^{\frac{18}{3}} = x^6$

Therefore, we have:

$\sqrt[3]{-27x^{18}} = -3x^6$

3. Final Answer

The simplified expression is $-3x^6$.