You have the expression:

$\left(\sqrt{-2}\right) \left(\sqrt{-24}\right)$

Let's simplify this step-by-step.

Step 1: Break down the square roots

Both square roots involve negative numbers, so we will work with complex numbers since $\sqrt{-1} = i$.

$\sqrt{-2} = \sqrt{2} \cdot i$ $\sqrt{-24} = \sqrt{24} \cdot i = \sqrt{4 \cdot 6} \cdot i = 2\sqrt{6} \cdot i$

Step 2: Multiply the expressions

Now we multiply the two expressions:

$\left(\sqrt{2} \cdot i\right) \cdot \left(2\sqrt{6} \cdot i\right)$

We first multiply the real parts and then the imaginary parts:

• Real part: $\sqrt{2} \cdot 2\sqrt{6} = 2\sqrt{12} = 2 \cdot 2\sqrt{3} = 4\sqrt{3}$
• Imaginary part: $i \cdot i = i^2 = -1$

Thus, the result is:

$4\sqrt{3} \cdot (-1) = -4\sqrt{3}$

Final Simplified Expression:

$-4\sqrt{3}$

Would you like more details on any step, or do you have any other questions?

Here are some related questions to further explore:

1. How do you simplify expressions involving square roots of negative numbers?
2. Can you explain how to handle complex numbers in square roots?
3. How do you multiply complex numbers like $i$ in an expression?
4. What is the general form for simplifying products of square roots?
5. How do you work with square roots of variables involving negatives?

Tip: When dealing with square roots of negative numbers, always remember to factor out $\sqrt{-1}$ as $i$ to simplify into complex numbers.