Question

Show me to understand and show work, please?

Answer 1

You have to multiply the two complex numbers, and simplify the result as one single complex number.

A complex number is composed by two terms: the real part, which is a real number, and an imaginary part, which is a multiple of i, the imaginary unit.

So, for example, the first factor is $15-4i$, which means that the real part is 15, and the imaginary part is -4i.

To multiply two complex numbers, you multiply each terms of the first number with each terms of the second number, just like you would multiply two polynomials like $(x+3)(x-2)$. The only exception is that you have to keep in mind that $i^2=-1$.

So, if we multiply these two numbers term by term we have

$(15-4i)(6-3i) = 15\cdot 6 - 15\cdot(-3i) + (-4i)\cdot 6 + (-4i)(-3i)$

This can be simplified to

$90 - 45i-24i+12i^2$

Summing like terms and recalling that $i^2=-1$ we have

$90-69i-12 = 78-69i$

So, the multiplication of the two factor gives as result the complex number $78-69i$. This means that the real part is 78, and the imaginary part is 69i. If you compare the two forms, you have

$78-69i = a+bi \iff a=78\quad b=-69$