It looks like you're working with the expression \( 4 \cdot 3 \sqrt{24x^4y^5} \). Let's simplify this step by step.

### Step 1: Break down the square root
We start with the expression inside the square root: \( 24x^4y^5 \). The goal is to simplify this.

- **Factor 24**: \( 24 = 4 \times 6 \), so \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2 \sqrt{6} \).
- **Simplify \( x^4 \)**: \( \sqrt{x^4} = x^2 \).
- **Simplify \( y^5 \)**: Split it as \( y^5 = y^4 \times y \), so \( \sqrt{y^5} = \sqrt{y^4 \times y} = \sqrt{y^4} \times \sqrt{y} = y^2 \sqrt{y} \).

So, \( \sqrt{24x^4y^5} = 2x^2 y^2 \sqrt{6y} \).

### Step 2: Multiply by 4 and 3
Now, multiply the simplified square root by \( 4 \times 3 \):
\[
4 \cdot 3 \sqrt{24x^4y^5} = 12 \times 2x^2 y^2 \sqrt{6y} = 24x^2 y^2 \sqrt{6y}.
\]

### Final Answer:
The simplified expression is:
\[
24x^2 y^2 \sqrt{6y}.
\]

Would you like any further details or explanations on this simplification?

Here are 5 related questions you can explore:
1. How do you simplify square roots with exponents?
2. What are the rules for multiplying square roots?
3. How do you handle square roots involving variables?
4. Can you simplify \( \sqrt{50x^6y^7} \) using similar steps?
5. How can you factor large numbers for easier simplification?

**Tip:** Always factor numbers and variables inside a square root to simplify them step by step.

Explore the step-by-step process to simplify the expression 4 * 3√24x^4y^5, covering algebraic concepts like radical simplification and exponent rules. Suitable for students in Grades 9-12, this guide explains how to break down square roots and simplify variables with exponents.

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