Final answer:
To divide the complex numbers (-7+9i) by (3-5i), multiply the numerator and denominator by the conjugate of the denominator (3+5i) and simplify. The resulting standard form of the complex number after simplification is 12/17 - 5i/17.
Step-by-step explanation:
The student is asked to perform the indicated operations involving complex numbers and then express the answer in standard form. The standard form of a complex number is a+bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, satisfying i2 = -1.
To perform the operation (-7+9i) / (3-5i), we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (3-5i) is (3+5i). Multiplying both parts by the conjugate we get:
((-7+9i)(3+5i)) / ((3-5i)(3+5i))
This results in:
(-21 + 35i - 45i - 45i2) / (9 - 15i + 15i - 25i2)
Since i2 = -1, we can simplify to:
(-21 - 10i + 45) / (9 + 25)
(24 - 10i) / 34
Dividing both real and imaginary parts by 34 results in:
(24/34) - (10i/34)
Simplifying we get:
12/17 - 5i/17 which is the answer in standard form.
To confirm the result, we should check the answer by ensuring that the denominator is not zero and the real and imaginary parts are appropriately simplified.