Math  /  Algebra

QuestionSimplify. (10+3i)(1717i)+(9i+13)-(-10+3 i)-(-17-17 i)+(-9 i+13)
Write your answer in the form a + bi. \square i

Studdy Solution

STEP 1

What is this asking? We're asked to simplify a complex expression with some negative signs and parentheses, and then write the answer in the standard *a + bi* form. Watch out! Be super careful with those negative signs; they can be tricky!
Don't rush, and double-check your work.

STEP 2

1. Distribute the negative signs
2. Group like terms
3. Simplify and write in standard form

STEP 3

Alright, let's **kick things off** by tackling those negative signs outside the parentheses!
Remember, a negative sign in front of parentheses means we multiply each term inside by 1-1.
It's like flipping the signs of everything inside!

STEP 4

So, distributing the negative signs, our expression becomes: (10+3i)(1717i)+(9i+13)=(103i)+(17+17i)+(9i+13)-(-10 + 3i) - (-17 - 17i) + (-9i + 13) = (10 - 3i) + (17 + 17i) + (-9i + 13). See how each term inside the first two sets of parentheses had its sign flipped?
That's the magic of distribution!

STEP 5

Now, let's **gather our friends**!
We'll group the real numbers (those without *i*) together and the imaginary numbers (those with *i*) together.
It's like sorting your socks – real with real, imaginary with imaginary!

STEP 6

Grouping like terms, we get: (10+17+13)+(3i+17i9i)(10 + 17 + 13) + (-3i + 17i - 9i). Notice how we've neatly arranged our real and imaginary terms, ready for the next step!

STEP 7

Time to **combine and conquer**!
Let's add the real numbers and then the imaginary numbers separately.

STEP 8

Adding the real parts, we have 10+17+13=4010 + 17 + 13 = 40.
That's our **real component**!

STEP 9

Adding the imaginary parts, we have 3i+17i9i=(3+179)i=5i-3i + 17i - 9i = (-3 + 17 - 9)i = 5i.
That's our **imaginary component**!

STEP 10

Finally, we put it all together in the standard *a + bi* form: 40+5i40 + 5i.
And there you have it – a beautifully simplified complex number!

STEP 11

Our final answer is 40+5i40 + 5i.

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