Math  /  Algebra

QuestionSimplify. (3i15)+(8i12)(-3 i-15)+(-8 i-12)
Write your answer in the form a + bi.

Studdy Solution

STEP 1

What is this asking? We're adding two complex numbers and want the answer in the standard *a* + *bi* form. Watch out! Don't forget that *i* is the imaginary unit, not just any variable!

STEP 2

1. Group like terms
2. Simplify

STEP 3

Alright, let's **group** those **real** and **imaginary** parts!
We've got (3i15)+(8i12)(-3i - 15) + (-8i - 12).
Think of it like sorting apples and oranges.
We'll put the real numbers (15-15 and 12-12) together and the imaginary numbers (3i-3i and 8i-8i) together.

STEP 4

So, rearranging, we get (1512)+(3i8i)(-15 - 12) + (-3i - 8i).
Remember, addition is commutative, so we can swap the order around without changing the sum!

STEP 5

Now, let's **combine** those **real numbers**: 1512=27-15 - 12 = -27.
Think of it like this: you owe someone $15\$15 and then you borrow $12\$12 more.
Now you owe a total of $27\$27, so that's 27-27.

STEP 6

Next, let's **tackle** the **imaginary parts**: 3i8i=11i-3i - 8i = -11i.
It's just like combining like terms with variables.
If you have 3-3 *i*'s and you subtract 8-8 more *i*'s, you end up with 11-11 *i*'s.

STEP 7

Putting it all together, we get 2711i-27 - 11i.
And there you have it!
Our answer is in the beautiful *a* + *bi* form, where *a* = 27-27 and *b* = 11-11.

STEP 8

2711i-27 - 11i

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