To express the complex number [tex]\(\frac{3 + 10i}{9 - i}\)[/tex] in the form [tex]\(a + bi\)[/tex], follow these steps:
1. Identify the given complex division:
[tex]\[
\frac{3 + 10i}{9 - i}
\][/tex]
2. Multiply the numerator and the denominator by the conjugate of the denominator:
The conjugate of [tex]\(9 - i\)[/tex] is [tex]\(9 + i\)[/tex].
Multiply both the numerator and the denominator by [tex]\(9 + i\)[/tex]:
[tex]\[
\frac{(3 + 10i)(9 + i)}{(9 - i)(9 + i)}
\][/tex]
3. Simplify the denominator:
The denominator is a product of a complex number and its conjugate:
[tex]\[
(9 - i)(9 + i) = 9^2 - (-i^2) = 81 + 1 = 82
\][/tex]
4. Expand and simplify the numerator:
Multiply each term in the numerator:
[tex]\[
(3 + 10i)(9 + i) = 3 \cdot 9 + 3 \cdot i + 10i \cdot 9 + 10i \cdot i = 27 + 3i + 90i + 10i^2
\][/tex]
Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
10i^2 = 10(-1) = -10
\][/tex]
Combining all terms:
[tex]\[
27 + 3i + 90i - 10 = 17 + 93i
\][/tex]
5. Express the resulting fraction:
[tex]\[
\frac{17 + 93i}{82}
\][/tex]
6. Separate the real and imaginary parts:
Divide both parts of the numerator by the denominator:
[tex]\[
\frac{17}{82} + \frac{93i}{82}
\][/tex]
7. Simplify the division:
Simplify each division to decimal form:
[tex]\[
\frac{17}{82} \approx 0.2073170731707317
\][/tex]
[tex]\[
\frac{93}{82} \approx 1.1341463414634148
\][/tex]
Therefore, the complex number [tex]\(\frac{3 + 10i}{9 - i}\)[/tex] in the form [tex]\(a + bi\)[/tex] is:
[tex]\[
0.2073170731707317 + 1.1341463414634148i
\][/tex]
So, the solutions are:
[tex]\[ a \approx 0.2073170731707317 \][/tex]
[tex]\[ b \approx 1.1341463414634148 \][/tex]
