Final answer:
To multiply (√3+i)⁸ times (1+i)⁵, first simplify (√3+i)⁸ and (1+i)⁵ using the properties of complex numbers. Then multiply these simplified expressions to find the final result. The correct option is d) 545 × 3^(1/2) - 418i.
Step-by-step explanation:
To perform the indicated operation (√3+i)⁸×(1+i)⁵, we can simplify using the properties of complex numbers:
(√3+i)⁸ = (3^(1/2) + i)⁸ = (3^(4/2) + 8 × (3^(1/2)) × i + 28 × i² + 56 × (3^(1/2)) × i³ + 70 × i⁴ + 56 × (3^(1/2)) × i⁵ + 28 × i⁶ + 8 × (3^(1/2)) × i⁷) = (3² + 8 × 3^(1/2) × i - 28 - 56 × 3^(1/2) × i + 70 + 56 × 3^(1/2) × i - 28 × i² + 8 × 3^(1/2) × (-i)) = (9 + 8 × 3^(1/2) × i - 28 - 56 × 3^(1/2) × i + 70 + 56 × 3^(1/2) × i + 28 - 8 × 3^(1/2) × i) = (-19 + 72 × 3^(1/2) × i)
(1+i)⁵ = (1 + i) × (1 + i)⁴ = (1 + i) × (1 + 4 × i + 6 × i² + 4 × i³ + i⁴) = (1 + i) × (1 + 4 × i + 6 × (-1) + 4 × (-i) + 1) = (1 + i) × (12 - i) = 12 + 12 × i - 1 × i - i² = 12 + 11 × i + 1 = 13 + 11 × i
Now, we can multiply these two simplified expressions:
(-19 + 72 × 3^(1/2) × i) × (13 + 11 × i) = (-19 × 13 + 72 × 3^(1/2) × i × 13 + 11 × i × (-19) + 11 × i × 72 × 3^(1/2)) = (-247 + 936 × 3^(1/2) × i - 209i + 792 × 3^(1/2) × i) = (-247 + 792 × 3^(1/2) × i - 209 × i + 936 × 3^(1/2) × i) = (-247 + 792 × 3^(1/2) × i - 209 × i + 936 × 3^(1/2) × i) = (-247 + 792 × 3^(1/2) × i - 209i + 936 × 3^(1/2) × i) = (545 × 3^(1/2) - 418i)
The result is 545 × 3^(1/2) - 418i. Therefore, the correct option is d) 545 × 3^(1/2) - 418i.