Final answer:
The reciprocal of the complex number 8-3i is found by dividing its complex conjugate (8+3i) by the modulus squared of 8-3i, which is 73. The result is 8/73 + 3i/73.
Step-by-step explanation:
The reciprocal of a complex number is the complex conjugate of that number divided by the modulus squared of the original number. In the case of the complex number 8-3i, its complex conjugate is 8+3i. To find the reciprocal of 8-3i, you would divide this complex conjugate by the modulus squared of 8-3i.
First, calculate the modulus squared of 8-3i:
- Modulus of 8-3i = sqrt(82 + (-3)2) = sqrt(64 + 9) = sqrt(73).
- Modulus squared = (sqrt(73))2 = 73.
Then, divide the complex conjugate by this modulus squared to get the reciprocal:
Reciprocal of 8-3i = (8+3i) / 73 = 8/73 + 3i/73.