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. Z=(V)/(I) If the voltage across a circuit element is 23+2i volts and the current through the element is 4-5i amperes, determine the impedance. The impedance is Z=square (Type your answer in the form a+bi

To determine the impedance \( Z \) of the circuit element, we need to divide the phasor voltage \( V \) by the phasor current \( I \). Given:<br /><br />\[ V = 23 + 2i \text{ volts} \]<br />\[ I = 4 - 5i \text{ amperes} \]<br /><br />The impedance \( Z \) is calculated as:<br /><br />\[ Z = \frac{V}{I} = \frac{23 + 2i}{4 - 5i} \]<br /><br />To simplify this expression, we multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \( 4 - 5i \) is \( 4 + 5i \).<br /><br />So, we have:<br /><br />\[ Z = \frac{(23 + 2i)(4 + 5i)}{(4 - 5i)(4 + 5i)} \]<br /><br />First, calculate the denominator:<br /><br />\[ (4 - 5i)(4 + 5i) = 4^2 - (5i)^2 = 16 - 25(-1) = 16 + 25 = 41 \]<br /><br />Next, calculate the numerator:<br /><br />\[ (23 + 2i)(4 + 5i) = 23 \cdot 4 + 23 \cdot 5i + 2i \cdot 4 + 2i \cdot 5i \]<br />\[ = 92 + 115i + 8i + 10i^2 \]<br />\[ = 92 + 123i + 10(-1) \]<br />\[ = 92 + 123i - 10 \]<br />\[ = 82 + 123i \]<br /><br />Now, divide the numerator by the denominator:<br /><br />\[ Z = \frac{82 + 123i}{41} = \frac{82}{41} + \frac{123i}{41} \]<br />\[ Z = 2 + 3i \]<br /><br />Therefore, the impedance \( Z \) is:<br /><br />\[ Z = 2 + 3i \text{ ohms} \]