Question

complex numbers are used to describe current, i, voltage, e, and impedance, z. these three quantities are related by the equation e = iz. given two of these quantities, solve the equation e = iz for the missing variable.
i = 8 + 8i, z = 5 + 2i
e = \boxed{}
(simplify your answer. type your answer in the form a + bi. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Substitute I and Z into E=IZ

We know \( I = 8 + 8i \) and \( Z = 5 + 2i \), so we substitute these into the equation \( E = IZ \). This gives us \( E=(8 + 8i)(5 + 2i) \).

Step2: Expand the product using FOIL method

Using the FOIL method (First, Outer, Inner, Last) for multiplying two binomials \((a + b)(c + d)=ac+ad+bc+bd\), we have:
First: \( 8\times5 = 40 \)
Outer: \( 8\times2i = 16i \)
Inner: \( 8i\times5 = 40i \)
Last: \( 8i\times2i = 16i^{2} \)
So, \( E = 40+16i + 40i+16i^{2} \)

Step3: Simplify using \( i^{2}=-1 \)

Since \( i^{2}=-1 \), we substitute \( i^{2} \) with -1 in the expression:
\( E = 40+16i + 40i+16\times(-1) \)
\( E = 40+16i + 40i - 16 \)

Step4: Combine like terms

Combine the real parts and the imaginary parts separately:
Real parts: \( 40-16 = 24 \)
Imaginary parts: \( 16i+40i = 56i \)
So, \( E = 24 + 56i \)

Answer:

\( 24 + 56i \)