The impedance​ Z, in​ ohms, of a circuit element is defined as the ratio of the phasor voltage​ V, in​ volts, across the element to the phasor current​ I, in​ amperes, through the elements. That​ is, Upper Z equals StartFraction Upper V Over Upper I EndFraction
. If the voltage across a circuit element is 20 plus 10 i volts and the current through the element is 4 minus 2 i ​amperes, determine the impedance.
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Part 1
The impedance is Zequals

enter your response here.
​(Type your answer in the form aplusbi​.)

To find the impedance \( Z \) of the circuit element, we use the given formula:

\[
Z = \frac{V}{I}
\]

where:
- \( V = 20 + 10i \) volts,
- \( I = 4 - 2i \) amperes.

The division of complex numbers \( Z = \frac{V}{I} \) can be carried out by multiplying the numerator and denominator by the conjugate of the denominator to simplify.

### Step-by-Step Solution
1. **Multiply the numerator and denominator by the conjugate of the denominator** (to remove the imaginary part in the denominator):
   \[
   Z = \frac{20 + 10i}{4 - 2i} \times \frac{4 + 2i}{4 + 2i}
   \]

2. **Simplify the numerator**:
   \[
   (20 + 10i)(4 + 2i) = 20 \cdot 4 + 20 \cdot 2i + 10i \cdot 4 + 10i \cdot 2i
   \]
   Expanding each term:
   \[
   = 80 + 40i + 40i + 20i^2
   \]
   Since \( i^2 = -1 \):
   \[
   = 80 + 80i - 20 = 60 + 80i
   \]

3. **Simplify the denominator**:
   \[
   (4 - 2i)(4 + 2i) = 4^2 - (2i)^2 = 16 - 4i^2
   \]
   Since \( i^2 = -1 \):
   \[
   = 16 + 4

The impedance​ Z, in​ ohms, of a circuit element is defined as the ratio of the phasor voltage​ V, in​ volts, across the element to the phasor current​ I, in​ amperes, through the element. That​ is, Z = V / I. If the voltage across a circuit element is 20 + 10i volts and the current through the element is 4 - 2i ​amperes, determine the impedance.

Learn to calculate the impedance of a circuit element by dividing complex numbers. Given voltage V = 20 + 10i volts and current I = 4 - 2i amperes, this solution demonstrates a step-by-step method using complex conjugate multiplication. Suitable for undergraduate students.

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