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