Final answer:
To simplify the expression √(-25 / (5 - 2i) + (1 - 31)), multiply the numerator and denominator by the conjugate of the denominator, simplify, and find the square root. The final answer is an imaginary number.
Step-by-step explanation:
To simplify the expression √(-25 / (5 - 2i) + (1 - 31)), we need to simplify the numerator and denominator separately before combining them.
Let's start with the numerator -25 / (5 - 2i). To simplify this, we can multiply the numerator and denominator by the conjugate of the denominator, which is (5 + 2i).
After simplifying the numerator and denominator, we can combine them and simplify further if possible.
The expression simplifies to √(15 - 8i - 155). Further simplification gives us √(-140 - 8i), which can be written as √(-140) * √(1 - i). The square root of -140 is an imaginary number, so the final answer is an imaginary number.