Final answer:
To simplify (8 + 3i) / (1 - 2i), multiply by the conjugate of the denominator, resulting in (2 + 19i) / 5. Hence, A = 2/5, B = 19/5, and C = 5.
Step-by-step explanation:
To simplify the complex fraction (8 + 3i) / (1 - 2i), we will multiply both the numerator and the denominator by the conjugate of the denominator to remove the imaginary unit 'i' from the denominator. The conjugate of 1 - 2i is 1 + 2i. By multiplying the numerator and denominator by this conjugate, we get:
(8 + 3i)(1 + 2i) / (1 - 2i)(1 + 2i)
Expanding both the numerator and the denominator yields:
(8 + 16i + 3i + 6i²) / (1 + 2i - 2i - 4i²)
Since i² = -1, the expression simplifies further to:
(8 + 19i - 6) / (1 + 4)
Which further simplifies to:
(2 + 19i) / 5
Thus, we have:
a = 2 / 5, b = 19 / 5, c = 5