Math  /  Algebra

QuestionSimplify. (0.1i+1.7)(6.18.6i)+(6.3+1.7i)(-0.1 i+1.7)-(-6.1-8.6 i)+(-6.3+1.7 i)
Write your answer in the form a+bi\mathrm{a}+\mathrm{bi}. \square

Studdy Solution

STEP 1

What is this asking? We need to simplify a complex number expression and write the answer in the standard form a+bia + bi, where aa is the real part and bb is the imaginary part. Watch out! Be careful with the signs when combining like terms!
It's easy to make a mistake if you rush.

STEP 2

1. Group Like Terms
2. Combine Like Terms

STEP 3

Let's rewrite our expression to group the real and imaginary parts together.
This makes it much easier to see what we need to combine! (0.1i+1.7)(6.18.6i)+(6.3+1.7i)(-0.1i + 1.7) - (-6.1 - 8.6i) + (-6.3 + 1.7i)

STEP 4

Remember, subtracting a negative is the same as adding a positive!
Distributing the negative sign gives us: (0.1i+1.7)+(6.1+8.6i)+(6.3+1.7i)(-0.1i + 1.7) + (6.1 + 8.6i) + (-6.3 + 1.7i)

STEP 5

Now, let's rearrange the terms to put the real parts together and the imaginary parts together.
This is like organizing your toys! (1.7+6.16.3)+(0.1i+8.6i+1.7i)(1.7 + 6.1 - 6.3) + (-0.1i + 8.6i + 1.7i)

STEP 6

Let's add the real numbers together: 1.7+6.16.31.7 + 6.1 - 6.3. 1.7+6.11.7 + 6.1 gives us **7.8**.
Then, 7.86.37.8 - 6.3 gives us **1.5**.
So, the real part of our simplified complex number is **1.5**.

STEP 7

Now, let's add the imaginary parts together: 0.1i+8.6i+1.7i-0.1i + 8.6i + 1.7i.
Adding 0.1i-0.1i and 8.6i8.6i gives us 8.5i8.5i.
Adding 8.5i8.5i to 1.7i1.7i gives us **10.2**i\mathrm{i}.
So, the imaginary part of our simplified complex number is **10.2**i\mathrm{i}.

STEP 8

Putting the real and imaginary parts together, we get our final answer: 1.5+10.2i1.5 + 10.2i.
Awesome!

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