Final answer:
To simplify the complex division 3 / (9 - 4i), multiply by the conjugate to eliminate the imaginary part in the denominator, resulting in the standard form 0.278 + 0.124i.
Step-by-step explanation:
To divide a complex number, such as 3 / (9 - 4i), by another complex number, we must multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 9 - 4i is 9 + 4i. Doing so will eliminate the imaginary part in the denominator and provide us with a real denominator which simplifies the expression. So, we get:
(3 / (9 - 4i)) × ((9 + 4i) / (9 + 4i)) = (3(9 + 4i)) / ((9 - 4i)(9 + 4i)).
Now, we perform the multiplication in the numerator and the foil method in the denominator:
Numerator: 3 × 9 + 3 × 4i = 27 + 12i
Denominator: (9 - 4i)(9 + 4i) = 9² - (4i)² = 81 - (-16) = 97.
So, our result is:
(27 + 12i) / 97 = 27/97 + (12/97)i,
which is the standard form of the complex quotient. Therefore, we have simplified the original complex division to 0.278 + 0.124i.