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.