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To solve the expression \(-7 \sqrt[3]{250} - 5 \sqrt[3]{16}\), we need to simplify each cube root term individually and then check if they can be combined.

### Step 1: Simplify \(\sqrt[3]{250}\)
First, break down 250 into its prime factors:
\[
250 = 2 \times 5^3
\]
Thus,
\[
\sqrt[3]{250} = \sqrt[3]{2 \times 5^3} = 5 \times \sqrt[3]{2}
\]
So, 
\[
-7 \sqrt[3]{250} = -7 \times 5 \times \sqrt[3]{2} = -35 \sqrt[3]{2}
\]

### Step 2: Simplify \(\sqrt[3]{16}\)
Next, break down 16 into its prime factors:
\[
16 = 2^4
\]
Thus,
\[
\sqrt[3]{16} = \sqrt[3]{2^4} = 2 \times \sqrt[3]{2}
\]
So,
\[
-5 \sqrt[3]{16} = -5 \times 2 \times \sqrt[3]{2} = -10 \sqrt[3]{2}
\]

### Step 3: Combine the Terms
Now that both terms are simplified, we have:
\[
-35 \sqrt[3]{2} - 10 \sqrt[3]{2}
\]
Since both terms contain \(\sqrt[3]{2}\), we can combine them:
\[
(-35 - 10) \sqrt[3]{2} = -45 \sqrt[3]{2}
\]

### Final Answer:
The simplified expression is:
\[
\boxed{-45 \sqrt[3]{2}}
\]

Do you want more details on this solution or have any other questions?

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Here are 5 related questions to expand on this concept:
1. How would you simplify the expression \(\sqrt[3]{54} + \sqrt[3]{128}\)?
2. What happens if you try to simplify \(\sqrt[3]{81} + 2\sqrt[3]{27}\)?
3. How do you simplify and combine the terms \(3\sqrt[3]{500} - 2\sqrt[3]{250}\)?
4. Can \(\sqrt[3]{250} + \sqrt[3]{32}\) be simplified further?
5. How do you approach simplifying radical expressions with higher powers, like \(\sqrt[4]{80}\)?

**Tip:** When simplifying cube roots, always try to factor the number under the root to check if any factors are perfect cubes.

Combine the terms if possible or enter DNE if they cannot be combined: -7∛250 - 5∛16

Learn how to simplify the expression -7∛250 - 5∛16 by breaking down cube roots into simpler factors. This problem covers key algebraic concepts including the simplification of cube roots, suitable for students in Grades 9-12.

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