Final answer:
The division of the complex numbers (3+i)/(2-3i) results in 3/13 + 11/13 i when multiplying by the conjugate of the denominator (2+3i) and simplifying.
Step-by-step explanation:
To divide the complex numbers (3+i)/(2-3i), we need to multiply by the conjugate of the denominator to remove the imaginary part from the denominator. The conjugate of (2-3i) is (2+3i). Now, multiply both the numerator and the denominator by this conjugate:
((3+i)(2+3i)) / ((2-3i)(2+3i))
Expanding the numerator gives us:
(3×2) + (3×3i) + (i×2) + (i×3i) = (6 + 9i + 2i - 3), since i^2 = -1.
So, the numerator simplifies to:
(6 - 3) + (9i + 2i) = 3 + 11i.
The denominator, when expanded, is:
(2×2) + (2×3i) - (3i×2) - (3i×3i) = 4 - 6i + 6i - 9i^2
Since i^2 = -1, this simplifies to:
4 + 9 = 13.
Therefore, the division yields:
(3 + 11i) / 13, which separates into:
3/13 + 11/13 i.
So the result of the division is 3/13 + 11/13 i.