Final answer:
To solve (5+5i)⁴/(−1−i)⁶, we simplify both the numerator and denominator separately and then divide them, resulting in a purely imaginary number, 0+312.5i, which is answer choice c.
Step-by-step explanation:
To perform the indicated operation: (5+5i)⁴/(−1−i)⁶, we must first simplify both the numerator and the denominator.
To simplify (5+5i)⁴: We use the binomial theorem or calculate directly (for simplicity, we could also recognize it as a binomial with equal real and imaginary parts where n equals 4).
(5+5i)² = 25+50i+25i² = 25+50i−25 (since i² = -1)
= 0+50i
So (5+5i)⁴ = (0+50i)² = 0² + 2 · 0 · 50i − 50² · i² = -2500i² = 2500 (since i² = -1).
To simplify (−1−i)⁶: We recognize that (−1−i) is similar to (5+5i) in that it has equal real and imaginary parts, but with a negative sign, where n equals 6).
(−1−i)² = 1+2i+i² = 1+2i−1 = 2i,
So (−1−i)⁶ = (2i)³ = 8i³ = 8i² · i = 8(−1)i = -8i
We can now perform the division:
2500 / -8i = -312.5i² = 312.5 (since i² = -1).
The answer is 0+312.5i which matches choice c. Performing the indicated operation gives us a purely imaginary number.