Question

Evaluate the expression 1-4i/6i and write the result in the form a+bi

Answer 1

To evaluate the expression \(\frac{1 - 4i}{6i}\) and write it in the form \(a + bi\), we can follow these steps:

1. Rewrite the expression:
\[
\frac{1 - 4i}{6i}
\]

2. To simplify the division by \(6i\), we can multiply the numerator and denominator by the conjugate of the denominator, which in this case is \(-i\):
\[
= \frac{(1 - 4i)(-i)}{(6i)(-i)}
\]

3. Calculate the denominator:
\[
(6i)(-i) = -6i^2 = -6(-1) = 6
\]

4. Now calculate the numerator:
\[
(1 - 4i)(-i) = 1 \cdot (-i) + (-4i)(-i) = -i + 4i^2 = -i + 4(-1) = -i - 4
\]
So the numerator becomes:
\[
-4 - i
\]

5. Combine the results:
\[
\frac{-4 - i}{6} = \frac{-4}{6} + \frac{-i}{6} = -\frac{2}{3} - \frac{1}{6}i
\]

Thus, the expression \(\frac{1 - 4i}{6i}\) simplified to the form \(a + bi\) is:
\[
-\frac{2}{3} - \frac{1}{6}i
\]

Final answer:
\[
a = -\frac{2}{3}, \quad b = -\frac{1}{6}
\]

So, the result is:
\[
-\frac{2}{3} - \frac{1}{6}i
\]