Question

Divide and express the result in standard form.
StartFraction 7 plus 3 i Over 3 minus 7 i EndFraction
Question content area bottom
Part 1
A.1
1
B.negative 1
negative 1
C.i
i
D.negative i
negative i

Answer 1

To divide the complex numbers \( \frac{7 + 3i}{3 - 7i} \) and express the result in standard form, we first multiply the numerator and the denominator by the conjugate of the denominator.

The conjugate of \( 3 - 7i \) is \( 3 + 7i \). We will multiply both the numerator and denominator by this conjugate:

\[
\frac{7 + 3i}{3 - 7i} \cdot \frac{3 + 7i}{3 + 7i} = \frac{(7 + 3i)(3 + 7i)}{(3 - 7i)(3 + 7i)}
\]

Calculating the denominator:

\[
(3 - 7i)(3 + 7i) = 3^2 - (7i)^2 = 9 - (-49) = 9 + 49 = 58
\]

Now calculating the numerator:

\[
(7 + 3i)(3 + 7i) = 7 \cdot 3 + 7 \cdot 7i + 3i \cdot 3 + 3i \cdot 7i
\]
\[
= 21 + 49i + 9i + 21i^2
\]
\[
= 21 + 49i + 9i - 21 \quad (\text{since } i^2 = -1)
\]
\[
= 0 + 58i = 58i
\]

Putting it all together, we have:

\[
\frac{58i}{58} = i
\]

Thus, the result in standard form is:

\[
\boxed{i}
\]

So the correct answer is C. \( i \).