Questions: Perform the indicated operation simplify. Hint: A simplified complex number will be of the form (a+bi). [ (-2+4 i)(4+4 i)= ]

Transcript text: Question 5 1 pt 1 Details Perform the indicated operation \& simplify. Hint: A simplified complex number will be of the form $a+b i$. \[ (-2+4 i)(4+4 i)= \] $\square$

Solution

Solution Steps

Step 1: Expand the product using the distributive property

We begin by expanding the product $(-2 + 4i)(4 + 4i)$ using the distributive property (also known as the FOIL method for binomials): \[ (-2 + 4i)(4 + 4i) = (-2)(4) + (-2)(4i) + (4i)(4) + (4i)(4i). \]

Step 2: Perform the multiplication

Now, we compute each term: \[ (-2)(4) = -8, \] \[ (-2)(4i) = -8i, \] \[ (4i)(4) = 16i, \] \[ (4i)(4i) = 16i^2. \]

Step 3: Simplify $i^2$

Recall that $i^2 = -1$. Substitute this into the expression: \[ 16i^2 = 16(-1) = -16. \]

Step 4: Combine like terms

Now, combine all the terms: \[ -8 - 8i + 16i - 16. \] Combine the real parts and the imaginary parts: \[ (-8 - 16) + (-8i + 16i) = -24 + 8i. \]