Final answer:
To divide complex numbers, we multiply both the numerator and denominator by the conjugate of the denominator. In this case, the division (4+5i)/(5-3i) is performed by multiplying the numerator and denominator by (5+3i). The expression simplifies to (5+37i)/34.
Step-by-step explanation:
To divide the complex number (4+5i) by (5-3i), we need to multiply both the numerator and the denominator by the conjugate of the denominator.
The conjugate of (5-3i) is (5+3i).
Hence, the division (4+5i)/(5-3i) is performed by multiplying the numerator and denominator by (5+3i).
When we multiply the numerator and denominator by (5+3i), we get:
(4+5i) * (5+3i) / (5-3i) * (5+3i)
Using the FOIL method, we can multiply the numerator and denominator as follows:
(4*5 + 4*3i + 5*5i + 5*3i) / (5*5 - 5*3i + 3i*5 - 3i*3i)
This simplifies to:
(20 + 12i + 25i + 15i^2) / (25 - 15i + 15i - 9i^2)
(20 + 37i + 15i^2) / (25 - 9i^2)
Since i^2 = -1, the expression further simplifies to:
(20 + 37i + 15(-1)) / (25 - 9(-1))
(20 + 37i - 15) / (25 + 9)
(5+37i)/34