The given expression is:

$\sqrt[3]{\frac{(x+3)^3}{(x-4)^6}}$

We are asked to simplify this cube root.

Step 1: Apply the property of cube roots

The cube root of a fraction can be simplified by applying the cube root to both the numerator and the denominator:

[ \sqrt[3]{\frac{(x+3)^3}{(x-4)^6}} = \frac{\sqrt[3]{(x+3)^3}}{\sqrt[3]{(x-4)^6}} ]

Step 2: Simplify the cube roots

Now, simplify the cube roots of the terms:

[ \frac{\sqrt[3]{(x+3)^3}}{\sqrt[3]{(x-4)^6}} = \frac{x+3}{(x-4)^2} ]

Final Simplified Expression:

$\frac{x+3}{(x-4)^2}$

This is the simplified form of the original expression.

Would you like any further details on this simplification, or do you have any questions?

Here are 5 related questions to expand on this concept:

1. How would the simplification differ if the root was not a cube but a square?
2. What happens if we change the powers in the numerator and denominator?
3. How do you simplify higher-order roots, like a fourth or fifth root?
4. How does this simplification relate to logarithmic properties?
5. What would happen if the denominator was raised to a negative power?

Tip: When simplifying roots, always look for opportunities to break down the powers and apply root properties directly to individual terms.