Question
Divide. Express your answer in standard form. \( \frac{-4 i}{1+i} \)
Ask by Cummings Best. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(-2 - 2i\)
Solution
To divide the complex numbers \(\frac{-4i}{1+i}\) and express the result in standard form \(a + bi\), follow these steps:
1. **Multiply the numerator and the denominator by the conjugate of the denominator:**
The conjugate of \(1 + i\) is \(1 - i\).
\[
\frac{-4i}{1+i} \times \frac{1-i}{1-i} = \frac{-4i(1-i)}{(1+i)(1-i)}
\]
2. **Simplify the denominator:**
\[
(1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2
\]
3. **Expand the numerator:**
\[
-4i(1-i) = -4i + 4i^2 = -4i + 4(-1) = -4i - 4
\]
4. **Combine the results:**
\[
\frac{-4i - 4}{2} = \frac{-4}{2} + \frac{-4i}{2} = -2 - 2i
\]
**Final Answer:**
\(-2 - 2i\)
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Simplify this solution Extra Insights
To divide the complex numbers, we multiply both the numerator and the denominator by the conjugate of the denominator, which in this case is \(1 - i\): \[ \frac{-4i}{1+i} \cdot \frac{1-i}{1-i} = \frac{-4i(1-i)}{(1+i)(1-i)} \] Calculating the denominator: \[ (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \] Now calculating the numerator: \[ -4i(1-i) = -4i + 4i^2 = -4i - 4 \quad (\text{since } i^2 = -1) \] Combining gives us: \[ -4 - 4i \] Thus, we have: \[ \frac{-4 - 4i}{2} = -2 - 2i \] Now expressing this in standard form, we have: \[ -2 - 2i \] So, the final answer is: \[ -2 - 2i \]
