Simplify  $5\sqrt[3]{108}\div\sqrt[3]{-50}$53√108÷3√−50​ .The simplified expression is .

1. Break Down the Problem

We need to simplify the expression $5\sqrt[3]{108} \div \sqrt[3]{-50}$.

2. Relevant Concepts

The property of cube roots states that: $\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}$ This will help us simplify this expression further.

3. Analysis and Detail

First, we can rewrite the expression using the property of cube roots: $5\sqrt[3]{108} \div \sqrt[3]{-50} = 5 \cdot \frac{\sqrt[3]{108}}{\sqrt[3]{-50}} = 5\sqrt[3]{\frac{108}{-50}}$

Now, we simplify the fraction inside the cube root: $\frac{108}{-50} = \frac{108}{50} \cdot (-1) = \frac{54}{25} \cdot (-1)$

Now, we can plug this back into the expression: $5\sqrt[3]{\frac{54}{25} \cdot (-1)} = 5\sqrt[3]{- \frac{54}{25}}$

4. Verify and Summarize

Next, we can separate the cube root: $5\sqrt[3]{-1} \cdot \sqrt[3]{\frac{54}{25}} = 5 \cdot (-1) \cdot \frac{\sqrt[3]{54}}{\sqrt[3]{25}} = -5 \cdot \frac{\sqrt[3]{54}}{5} = -\sqrt[3]{54}$

Final Answer

The simplified expression is: $-\sqrt[3]{54}$