QuestionSimplify $i(2+4 i)(-3 i)$ and express it as $a + bi$ .

Studdy Solution

STEP 1

Assumptions1. The expression is $i(+4 i)(0+(-3 i))$ . We need to simplify the expression and write it in the form $a+bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit with the property $i^ = -1$ .

STEP 2

First, we distribute the $i$ to the terms inside the first parentheses.
$i(2+4 i) =2i +4i^2$

STEP 3

Now, we replace $i^2$ with $-1$ .
$2i +i^2 =2i +(-1) =2i -$

STEP 4

Next, we rearrange the terms to match the form $a+bi$ .
$2i -4 = -4 +2i$

STEP 5

Now, we multiply the expression $-4 +2i$ with the second parentheses $(0+(-3 i))$ .
$(-4 +2i)(0+(-3 i)) = -4(0) + -4(-3i) +2i(0) +2i(-3i)$

STEP 6

implify the expression by performing the multiplications.
$-4(0) + -4(-3i) +2i(0) +2i(-3i) =0 +12i +0 -6i^2$

STEP 7

Replace $i^2$ with $-1$ .
$0 +12i +0 -6i^2 =0 +12i -6(-1) =12i +6$

STEP 8

Rearrange the terms to match the form $a+bi$ .
$12i +6 =6 +12i$ So, the simplified form of the expression $i(2+4 i)(0+(-3 i))$ is $6 +12i$ .

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