Evaluating Complex Expressions: (5 + i)^2 / (3  i)
This article will guide you through the process of evaluating the complex expression (5 + i)^2 / (3  i). We will use fundamental operations with complex numbers to arrive at the final simplified form.
Understanding Complex Numbers
A complex number is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of 1 (i.e., i^2 = 1).
Simplifying the Expression

Expanding the Square: Begin by expanding the square in the numerator: (5 + i)^2 = (5 + i)(5 + i) = 25 + 5i + 5i + i^2 Since i^2 = 1, we have: (5 + i)^2 = 25 + 10i  1 = 24 + 10i

Multiplying by the Conjugate: To simplify the division, we multiply both the numerator and denominator by the conjugate of the denominator (3  i). The conjugate of a complex number is obtained by changing the sign of the imaginary part. The conjugate of (3  i) is (3 + i).
Therefore, we get: (24 + 10i) / (3  i) * (3 + i) / (3 + i)

Expanding and Simplifying: Expanding the numerator and denominator: (24 + 10i)(3 + i) / (3  i)(3 + i) = (72 + 24i + 30i + 10i^2) / (9  i^2) Simplifying using i^2 = 1: (72 + 54i  10) / (9 + 1) = (62 + 54i) / 10

Final Simplified Form: Finally, we can express the result in the standard form a + bi: (62 + 54i) / 10 = 6.2 + 5.4i
Conclusion
Therefore, the simplified form of the complex expression (5 + i)^2 / (3  i) is 6.2 + 5.4i. This process demonstrates how to work with complex numbers, using basic arithmetic operations, conjugate multiplication, and simplification techniques.