We are tasked with evaluating the expression:

$V = \frac{1}{\sqrt{-1} \cdot \left(0.5 \cdot 10^{-12}\right) \cdot \left(2.4 \cdot 10^9\right)} + \sqrt{-1} \cdot \left(2.5 \cdot 10^{-9}\right) \cdot \left(2.4 \cdot 10^9\right)$

Breaking it down:

1. The term $\sqrt{-1}$ represents the imaginary unit, commonly written as $i$.
2. We can substitute $\sqrt{-1} = i$ into the equation to simplify.

Thus, we rewrite the expression as:

$V = \frac{1}{i \cdot (0.5 \cdot 10^{-12}) \cdot (2.4 \cdot 10^9)} + i \cdot (2.5 \cdot 10^{-9}) \cdot (2.4 \cdot 10^9)$

First Term:

Simplify the denominator of the first term:

$i \cdot (0.5 \cdot 10^{-12}) \cdot (2.4 \cdot 10^9) = i \cdot (0.5 \cdot 2.4) \cdot 10^{-3} = i \cdot 1.2 \cdot 10^{-3}$

Now, the first term is:

$\frac{1}{i \cdot 1.2 \cdot 10^{-3}} = \frac{-i}{1.2 \cdot 10^{-3}} = -\frac{i}{1.2 \cdot 10^{-3}} = -833.33i$

Second Term:

For the second term:

$i \cdot (2.5 \cdot 10^{-9}) \cdot (2.4 \cdot 10^9) = i \cdot (2.5 \cdot 2.4) = i \cdot 6 = 6i$

Combining Terms:

Now, we combine both terms:

$V = -833.33i + 6i = (-833.33 + 6)i = -827.33i$

Thus, the value of $V$ is:

$V = -827.33i$

---

Would you like further explanation or have any additional questions on this? Here are a few questions that expand on this:

1. How does the imaginary unit $i$ affect the magnitude and phase of a complex number?
2. What are some applications of complex numbers in real-world physics or engineering problems?
3. How can we express a complex number in polar form?
4. What happens when you multiply two complex numbers, and how do we calculate the product?
5. Could you explain the concept of impedance in AC circuits and how complex numbers are used there?

Tip: When dealing with complex numbers, always remember that $i^2 = -1$, which is fundamental for simplifying expressions involving powers of $i$.