Question

Divide and simplify to the form aplusbi.
StartFraction 20 i Over 3 plus i EndFraction

Answer 1

To simplify the expression \(\frac{20i}{3 + i}\) into the form \(a + bi\), we can multiply the numerator and the denominator by the conjugate of the denominator.

The conjugate of \(3 + i\) is \(3 - i\). So, we will multiply both the numerator and the denominator by \(3 - i\):

\[
\frac{20i}{3 + i} \cdot \frac{3 - i}{3 - i} = \frac{20i(3 - i)}{(3 + i)(3 - i)}
\]

Now, let's simplify the denominator:

\[
(3 + i)(3 - i) = 3^2 - i^2 = 9 - (-1) = 10
\]

Next, let's simplify the numerator:

\[
20i(3 - i) = 60i - 20i^2
\]

Since \(i^2 = -1\), we have:

\[
-20i^2 = -20(-1) = 20
\]

Now, combining terms in the numerator gives us:

\[
60i + 20
\]

So now, we can put it all together:

\[
\frac{60i + 20}{10} = \frac{20}{10} + \frac{60i}{10} = 2 + 6i
\]

Therefore, the simplified result in the form \(a + bi\) is:

\[
\boxed{2 + 6i}
\]