We are given the formula for Ohm's law:

$E = I \times R$

where:

• $E$ is the voltage in volts,
• $I$ is the current in amperes,
• $R$ is the resistance in ohms.

We are provided the following complex values:

• $I = 6 + 4i$ amperes,
• $R = 6 + 9i$ ohms.

To find $E$, we will multiply the complex numbers $I$ and $R$.

Step 1: Multiply the complex numbers

$E = (6 + 4i) \times (6 + 9i)$

Using the distributive property (FOIL method for complex numbers):

$E = (6 \times 6) + (6 \times 9i) + (4i \times 6) + (4i \times 9i)$ $E = 36 + 54i + 24i + 36i^2$

Step 2: Simplify

Since $i^2 = -1$, we can replace $i^2$ with $-1$:

$E = 36 + 54i + 24i + 36(-1)$ $E = 36 + 54i + 24i - 36$ $E = (36 - 36) + (54i + 24i)$ $E = 0 + 78i$

Thus, the voltage $E$ is $0 + 78i$ volts.

Final Answer:

The correct answer is:

A. $(0 + 78i)$ volts

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Would you like more details on any of the steps, or do you have additional questions?

Here are some related questions:

1. How do you multiply complex numbers using the FOIL method?
2. What is the significance of the imaginary unit $i$ in complex numbers?
3. Can Ohm's law be applied to circuits with real-only values for current and resistance?
4. How can complex impedance be used to describe AC circuits in electronics?
5. What happens if the resistance or current in a circuit is purely imaginary?

Tip: When working with complex numbers in physics, always remember that $i^2 = -1$. This is crucial when simplifying expressions involving complex numbers.