Let's simplify the expression:

$\left( \sqrt{-2} \right) \left( 8\sqrt{-7} \right)$

Step 1: Simplify the square roots involving negative numbers.

• $\sqrt{-2}$ can be written as $\sqrt{2} \times i$, where $i$ is the imaginary unit, so $\sqrt{-2} = i\sqrt{2}$.
• $\sqrt{-7}$ can be written as $\sqrt{7} \times i$, so $\sqrt{-7} = i\sqrt{7}$.

Thus, the expression becomes:

$\left( i\sqrt{2} \right) \left( 8i\sqrt{7} \right)$

Step 2: Multiply the expressions.

Now, multiply the two expressions:

$i\sqrt{2} \times 8i\sqrt{7}$

First, multiply the coefficients:

$i \times 8i = 8i^2$

Next, multiply the square root terms:

$\sqrt{2} \times \sqrt{7} = \sqrt{14}$

So the expression is:

$8i^2 \times \sqrt{14}$

Step 3: Simplify using $i^2 = -1$.

Since $i^2 = -1$, the expression becomes:

$8(-1)\sqrt{14} = -8\sqrt{14}$

Final Answer:

The fully simplified expression is:

$\boxed{-8\sqrt{14}}$

Would you like to explore any steps further or have any related questions? Here are some related questions:

1. How do you simplify expressions involving imaginary numbers?
2. What is the geometric interpretation of multiplying two complex numbers?
3. How do you handle expressions with multiple square roots and imaginary units?
4. What are the rules for simplifying expressions with $i^2$?
5. How do you multiply and divide radicals in algebra?

Tip: Remember that $i^2 = -1$ is a key property when working with imaginary numbers.