3Sqrt-24x^10yY^8

To simplify the expression 3√(-24x^10yY^8), we can break it down step by step.

1. First, let's identify the perfect square factors in the expression. In this case, the expression can be written as 3√(-1 * 2^3 * x^10 * y * Y^8).

2. Next, we can group the perfect square factors together, which are -1, 2^2, x^5, Y^4. The remaining factors can be considered as a single factor.

3. Now, let's simplify the perfect square factors:
- The square root of -1 is represented by the imaginary unit "i".
- The square root of 2^2 is 2.
- The square root of x^5 is x^(5/2) since we can divide the exponent by 2.
- The square root of Y^4 is Y^(4/2) which simplifies to Y^2.

4. Putting it all together, we have 3 * i * 2 * x^(5/2) * Y^2 * √(yY^8).

So the simplified expression is 6ix^(5/2)Y^2√(yY^8).