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The expression you provided is:

\[
\sqrt[3]{6x^3 y^7} \cdot \sqrt[3]{4x^5}
\]

### Simplifying the expression:

We will apply the property of cube roots that allows us to combine them under one radical when multiplied:

\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\]

Thus, we can rewrite the expression as:

\[
\sqrt[3]{(6x^3 y^7) \cdot (4x^5)}
\]

### Multiply inside the radical:

\[
6x^3 y^7 \cdot 4x^5 = (6 \cdot 4) \cdot (x^3 \cdot x^5) \cdot y^7 = 24x^8 y^7
\]

Now we have:

\[
\sqrt[3]{24x^8 y^7}
\]

### Simplifying the cube root:

Now, let's extract cubes from under the radical:

1. \( x^8 = x^{3 \cdot 2 + 2} = (x^6) \cdot x^2 \), so we can take out \( x^2 \) since \( x^6 \) is a perfect cube.
2. \( y^7 = y^{3 \cdot 2 + 1} = (y^6) \cdot y \), so we can take out \( y^2 \) since \( y^6 \) is a perfect cube.

Thus, the expression becomes:

\[
x^2 y^2 \sqrt[3]{24 x^2 y}
\]

This is the simplified form of the given expression.

Would you like further details or have any questions?

### Related Questions:
1. How do you simplify expressions with higher roots (e.g., fourth or fifth roots)?
2. Can you explain how to handle fractional exponents when simplifying radicals?
3. What other properties of radicals can help in simplifying complex expressions?
4. How can you use prime factorization to simplify cube roots?
5. What are the applications of radical simplification in real-world problems?

**Tip:** When simplifying radicals, always look for perfect squares, cubes, etc., within the expression to make it easier to extract values from the radical.

Simplify the expression: ∛(6x^3 y^7) · ∛(4x^5)

Learn how to simplify the cube root expression ∛(6x^3 y^7) · ∛(4x^5) by applying the properties of radicals and the laws of exponents. This problem is suitable for students in Grades 9-12.

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