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Math  /  Algebra

QuestionSimplify. −(1+8i2)+6−5i2-\left(\frac{1+8 i}{2}\right)+\frac{6-5 i}{2}
Write your answer in the form a + bi. Simplify all fractions. □\square i

Studdy Solution

STEP 1

What is this asking? We've got two complex numbers, one is being subtracted from the other, and we need to combine them and write the result in the standard *a* + *bi* form. Watch out! Don't forget to distribute that negative sign to both parts of the first complex number!

STEP 2

1. Distribute the negative sign.
2. Combine the real parts.
3. Combine the imaginary parts.
4. Write the final answer.

STEP 3

Alright, let's **kick things off** by distributing that negative sign to both the real and imaginary parts inside the parentheses.
Remember, distributing a negative is like multiplying by −1-1.
So, we're doing this because the problem asks us to subtract the first complex number from the second.
−(1+8i2)=−1⋅(1+8i)2=−1−8i2-\left(\frac{1+8i}{2}\right) = \frac{-1 \cdot (1+8i)}{2} = \frac{-1 - 8i}{2}

STEP 4

Now, let's **wrangle those real parts**!
We've got −1/2-1/2 from our first complex number and 6/26/2 from our second.
We're adding these together because we're combining the two complex numbers.
−12+62=−1+62=52\frac{-1}{2} + \frac{6}{2} = \frac{-1 + 6}{2} = \frac{5}{2}

STEP 5

Time to **tackle the imaginary parts**!
We have −8i/2-8i/2 and −5i/2-5i/2.
Remember, we're adding these together because the original problem is adding the two complex numbers (after we distributed the negative sign).
−8i2+−5i2=−8i−5i2=−13i2\frac{-8i}{2} + \frac{-5i}{2} = \frac{-8i - 5i}{2} = \frac{-13i}{2}

STEP 6

Let's **put it all together** now!
We've got our real part, 5/25/2, and our imaginary part, −13i/2-13i/2.
Remember, the standard form is *a* + *bi*.
52−132i\frac{5}{2} - \frac{13}{2}i

STEP 7

52−132i\frac{5}{2} - \frac{13}{2}i

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