Complex numbers are used to describe​ current, I,​ voltage, E, and​ impedance, Z. These three quantities are related by the equation =IZ. Given two of these​ quantities, solve the equation E=IZ for the missing variable.

I=7+ 5i​, E=4 + 8i 

To solve the equation \( E = IZ \) for the missing variable \( Z \), given the values of \( I \) and \( E \), follow these steps:

### Given:
- \( I = 7 + 5i \) (current)
- \( E = 4 + 8i \) (voltage)

### Equation to solve:
\[ E = IZ \]

### Step 1: Isolate \( Z \)
\[ Z = \frac{E}{I} \]

### Step 2: Substitute the given values into the equation
\[ Z = \frac{4 + 8i}{7 + 5i} \]

### Step 3: Multiply the numerator and the denominator by the conjugate of the denominator

The conjugate of \( 7 + 5i \) is \( 7 - 5i \). So, multiply both the numerator and the denominator by \( 7 - 5i \):

\[
Z = \frac{(4 + 8i)(7 - 5i)}{(7 + 5i)(7 - 5i)}
\]

### Step 4: Expand the numerator and the denominator

- **Numerator**:
  \[
  (4 + 8i)(7 - 5i) = 4(7) - 4(5i) + 8i(7) - 8i(5i)
  \]
  Simplifying this:
  \[
  28 - 20i + 56i - 40i^2 = 28 + 36i - 40(-1) = 28 + 36i + 40 = 68 + 36i
  \]

- **Denominator**:
  \[
  (7 + 5i)(7 - 5i) = 7^2 - (5i)^2 = 49 - 25(-1) = 49 + 25 = 74
  \]

### Step 5: Write the final expression for \( Z \)
\[
Z = \frac{68 + 36i}{74} = \frac{68}{74} + \frac{36i}{74} = \frac{34}{37} + \frac{18i}{37}
\]

So, the impedance \( Z \) is:
\[
\boxed{Z = \frac{34}{37} + \frac{18i}{37}}
\]

Would you like further details or have any additional questions?

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Here are 5 related questions to further explore the topic:

1. How can you interpret the real and imaginary parts of impedance in physical terms?
2. What would be the effect on \( Z \) if \( I \) and \( E \) were purely real numbers?
3. How does the concept of phasors relate to complex numbers in electrical engineering?
4. Can you calculate the magnitude and phase angle of the impedance \( Z \)?
5. How would you solve for \( I \) if \( Z \) and \( E \) were given?

**Tip:** Remember that in AC circuits, the real part of the impedance represents resistance, while the imaginary part represents reactance.

Learn how to solve the equation E = IZ for impedance Z using complex numbers. Follow a detailed step-by-step solution with examples. Suitable for college-level students and professionals in electrical engineering.

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