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

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.

Simplify the expression: \sqrt[3]{\frac{(x+3)^3}{(x-4)^6}}

Learn how to simplify the expression \sqrt[3]{\frac{(x+3)^3}{(x-4)^6}} step-by-step. This guide covers the cube root property and exponent rules for simplifying algebraic expressions. Perfect for high school students in Grades 9-12.

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