Question

Simplify the following expression using the order of operations and write it in the form of a + bi : (-2 + 7i) (3 + 8i) - (3 - 4i).65 + 9i65 - 9i-65 + 9i-65 - 9i

Answer 1

Step 1. The expression with complex numbers that we have is:

$(-2+7i)(3+8i)-(3-4i)$

We have multiplication and subtraction. First, we will need to solve the multiplications and after that, we will make the subtraction.

Step 2. The formula to multiply complex numbers is:

$(a+bi)(c+di)=(ac-bd)+(bc+ad)i$

In our multiplication:

a=-2

b=7

c=3

d=8

Therefore, the result of the first part of the expression is:

$(-2+7i)(3+8i)=(-2(3)-7(8))+(7(3)+(-2)(8))i$

Solving the operations:

$\begin{gathered} (-2+7\imaginaryI)(3+8\imaginaryI)=(-6-56)+(21-16)\imaginaryI \\ \downarrow \\ (-2+7\imaginaryI)(3+8\imaginaryI)=-62+5\imaginaryI \end{gathered}$

Substituting this result into the original expression:

$\begin{gathered} (-2+7\imaginaryI)(3+8\imaginaryI)-(3-4\imaginaryI) \\ \downarrow \\ (-62+5\mathrm{i})-(3-4\mathrm{i}) \end{gathered}$

Step 3. Now we need to make the subtraction. To subtract complex numbers, we use the following formula:

$(a+bi)-(c+di)=(a-c)+(b-d)i$

In our case:

a=-62

b=5

c=3

d=-4

The result is:

$(-62+5\imaginaryI)-(3-4\imaginaryI)=(-62-3)+(5-(-4))i$

Solving the operations:

$\begin{gathered} (-62+5\imaginaryI)-(3-4\imaginaryI)=(-65)+(5+4)\imaginaryI \\ \downarrow \\ (-62+5\imaginaryI)-(3-4\imaginaryI)=\boxed{-65+9\mathrm{i}} \end{gathered}$

Answer:

$\boxed{-65+9\mathrm{\imaginaryI}}$