QuestionEvaluate the product and express it as $a + bi$ : $(2 + 4i)^{2}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to evaluate the square of a complex number in the form (a+bi)^. . The complex number given is $+4i$ .
3. The product will be in the form $a+bi$ .

STEP 2

We can calculate the square of a complex number by multiplying the complex number by itself.
$(2+4i)^2 = (2+4i)(2+4i)$

STEP 3

Now, we will use the distributive property to multiply each term in the first complex number by each term in the second complex number.
$(2+i)(2+i) =2(2) +2(i) +i(2) +i(i)$

STEP 4

implify the multiplication.
$=4 +8i +8i +16i^2$

STEP 5

Remember that $i^2 = -1$ . Substitute $-1$ for $i^2$ .
$=4 +8i +8i -16$

STEP 6

Combine like terms.
$= -12 +16i$ So, the product of $(2+4i)^2$ is $-12+16i$ .

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