Question

Divide and simplify
135+25i/ 7+9i=

Answer 1

To divide the complex numbers \(\frac{135 + 25i}{7 + 9i}\), we can multiply the numerator and denominator by the conjugate of the denominator, which is \(7 - 9i\):

\[
\frac{135 + 25i}{7 + 9i} \cdot \frac{7 - 9i}{7 - 9i}
\]

Now, we will calculate the numerator and denominator separately.

**Numerator:**

\[
(135 + 25i)(7 - 9i) = 135 \cdot 7 + 135 \cdot (-9i) + 25i \cdot 7 + 25i \cdot (-9i)
\]
\[
= 945 - 1215i + 175i - 225i^2
\]
Since \(i^2 = -1\), we can replace \(-225i^2\) with \(225\):

\[
= 945 - 1215i + 175i + 225
\]
\[
= (945 + 225) + (-1215i + 175i)
\]
\[
= 1170 - 1040i
\]

**Denominator:**

\[
(7 + 9i)(7 - 9i) = 7^2 - (9i)^2 = 49 - 81(-1) = 49 + 81 = 130
\]

Now we can combine our results:

\[
\frac{1170 - 1040i}{130}
\]

Next, we can simplify by dividing both the real and imaginary parts by \(130\):

\[
= \frac{1170}{130} - \frac{1040i}{130}
\]

Calculating the division:

\[
= 9 - \frac{1040}{130}i
\]

Now, simplifying \(\frac{1040}{130}\):

\(\frac{1040}{130} = 8\)

Thus, the final result is:

\[
9 - 8i
\]

So, the simplified result is:

\[
\boxed{9 - 8i}
\]