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 , 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 . The only exception is that you have to keep in mind that .

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

This can be simplified to

Summing like terms and recalling that we have

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

Explanation:

Answer 2

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`