https://r2.erweima.ai/mathbot/image/1727037206345du6dwqzr.jpg

The expression in the image is:

\[
\left( \sqrt[3]{9} + \sqrt[3]{6} + \sqrt[3]{4} \right) \left( \sqrt[3]{3} - \sqrt[3]{2} \right)
\]

We can approach this by using distribution (expanding the product) between the two terms inside the parentheses. Let me first break it down:

### Step 1: Expand the expression
\[
\left( \sqrt[3]{9} + \sqrt[3]{6} + \sqrt[3]{4} \right)\left( \sqrt[3]{3} - \sqrt[3]{2} \right)
\]

Using distributive property, expand:

\[
= \sqrt[3]{9} \cdot \sqrt[3]{3} - \sqrt[3]{9} \cdot \sqrt[3]{2} + \sqrt[3]{6} \cdot \sqrt[3]{3} - \sqrt[3]{6} \cdot \sqrt[3]{2} + \sqrt[3]{4} \cdot \sqrt[3]{3} - \sqrt[3]{4} \cdot \sqrt[3]{2}
\]

Now let's simplify each term individually.

### Step 2: Simplify each product

- \( \sqrt[3]{9} \cdot \sqrt[3]{3} = \sqrt[3]{27} = 3 \)
- \( \sqrt[3]{9} \cdot \sqrt[3]{2} = \sqrt[3]{18} \)
- \( \sqrt[3]{6} \cdot \sqrt[3]{3} = \sqrt[3]{18} \)
- \( \sqrt[3]{6} \cdot \sqrt[3]{2} = \sqrt[3]{12} \)
- \( \sqrt[3]{4} \cdot \sqrt[3]{3} = \sqrt[3]{12} \)
- \( \sqrt[3]{4} \cdot \sqrt[3]{2} = \sqrt[3]{8} = 2 \)

### Step 3: Combine the simplified terms
\[
= 3 - \sqrt[3]{18} + \sqrt[3]{18} - \sqrt[3]{12} + \sqrt[3]{12} - 2
\]

Notice that \( - \sqrt[3]{18} \) and \( + \sqrt[3]{18} \) cancel out, and \( - \sqrt[3]{12} \) and \( + \sqrt[3]{12} \) cancel out as well, leaving:

\[
= 3 - 2 = 1
\]

### Final Answer:
The expression simplifies to \( 1 \).

---

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How does the distributive property work for cube roots in general?
2. What are the steps to simplify cube roots?
3. Can you provide an example where cube roots lead to a more complex expression?
4. How can we handle expressions involving higher-order roots (like fourth roots)?
5. Can this type of simplification be extended to larger polynomial products?

**Tip:** When working with cube roots, try simplifying each term first before combining, as many terms may cancel out!

Learn how to simplify the cube roots expression (∛9 + ∛6 + ∛4)(∛3 - ∛2) step by step. This problem uses the distributive property and simplifies cube roots, ideal for high school students in Grades 9-12.

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