To simplify the complex fraction [tex]\(\frac{3-4i}{9+4i}\)[/tex], we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(9+4i\)[/tex] is [tex]\(9-4i\)[/tex].
Here's the detailed step-by-step solution:
1. Identify the complex fraction to be simplified:
[tex]\[
\frac{3-4i}{9+4i}
\][/tex]
2. Multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{(3-4i)(9-4i)}{(9+4i)(9-4i)}
\][/tex]
3. Use the formula for multiplying complex numbers:
[tex]\[
(a+bi)(c+di) = (ac-bd) + (ad+bc)i
\][/tex]
4. For the numerator [tex]\((3-4i)(9-4i)\)[/tex]:
[tex]\[
3 \cdot 9 + 3 \cdot (-4i) - 4i \cdot 9 - 4i \cdot (-4i)
\][/tex]
[tex]\[
= 27 - 12i - 36i + 16i^2
\][/tex]
Since [tex]\( i^2 = -1 \)[/tex], we have:
[tex]\[
= 27 - 12i - 36i + 16(-1)
\][/tex]
[tex]\[
= 27 - 12i - 36i - 16
\][/tex]
[tex]\[
= 11 - 48i
\][/tex]
5. For the denominator [tex]\((9+4i)(9-4i)\)[/tex]:
[tex]\[
9 \cdot 9 + 9 \cdot (-4i) + 4i \cdot 9 - 4i \cdot 4i
\][/tex]
[tex]\[
= 81 - 36i + 36i - 16i^2
\][/tex]
Since [tex]\( i^2 = -1 \)[/tex], we have:
[tex]\[
= 81 - 16(-1)
\][/tex]
[tex]\[
= 81 + 16
\][/tex]
[tex]\[
= 97
\][/tex]
6. Combine the results:
[tex]\[
\frac{11 - 48i}{97}
\][/tex]
7. Separate the real and imaginary parts:
[tex]\[
\frac{11}{97} - \frac{48}{97}i
\][/tex]
Thus, the simplified form of the given complex fraction is:
[tex]\[
\frac{11}{97} - \frac{48}{97}i
\][/tex]
From the provided options, the correct answer is:
c. [tex]\(\frac{11}{97} - \frac{48}{97} i \)[/tex]
