We are tasked with calculating the value of the following expression:

$\sqrt{2i} \times \sqrt{6i}$

Step 1: Combine the square roots

Using the property of square roots that says $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$, we can combine the square roots:

$\sqrt{2i} \times \sqrt{6i} = \sqrt{(2i) \times (6i)}$

Step 2: Simplify the product inside the square root

Now, multiply the two complex numbers inside the square root:

$(2i) \times (6i) = 12i^2$

Since $i^2 = -1$, we have:

$12i^2 = 12(-1) = -12$

So, the expression becomes:

$\sqrt{-12}$

Step 3: Simplify the square root

Now, $\sqrt{-12}$ can be simplified as:

$\sqrt{-12} = \sqrt{12} \times \sqrt{-1}$

Since $\sqrt{-1} = i$, we get:

$\sqrt{12} \times i$

We can further simplify $\sqrt{12}$:

$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$

Thus, we have:

$\sqrt{-12} = 2\sqrt{3} \times i$

Final Answer:

The simplified value of the expression is:

$2i\sqrt{3}$

Would you like more details on any step?

Related questions:

1. How can we multiply two complex numbers using polar form?
2. What are the general rules for simplifying square roots of complex numbers?
3. How does the imaginary unit $i$ behave in higher powers?
4. How do you express complex numbers in exponential form (Euler's formula)?
5. What is the significance of complex conjugates in simplifying expressions like this?

Tip: Always remember that $i^2 = -1$ and this helps simplify many square roots involving complex numbers!