The value of ( (- 1 + i√3)/(1 - i) )³⁰ is :

Solution

Maths sir · 4 steps

Turn the top and bottom into polar form first, then the 30th power is only a rotation.

1. Polar form. The numerator -1 + i√3 has modulus 2 and lies in the second quadrant, so its argument is 2π/3. The denominator 1 - i has modulus √2 and lies in the fourth quadrant, so its argument is -π/4.
2. Divide. Divide the moduli and subtract the arguments: modulus 2/√2 = √2 and argument 2π/3 - (-π/4) = 11π/12.

   (√2 e^(i 11π/12))³⁰ = (√2)³⁰ e^(i 330π/12) = 2¹⁵ e^(i 55π/2)

3. Reduce the angle. 55π/2 = 26π + 3π/2, and 26π is 13 full turns. So the angle is 3π/2, and cos 3π/2 + i sin 3π/2 = -i.
4. So. The value is 2¹⁵ × (-i) = -2¹⁵i, which is option (a).

The idea to keep. For a big power of a quotient, find each modulus and argument, divide and subtract, then use De Moivre. Remove full turns of 2π from the final angle before reading off cos and sin.

Answer from the NTA final answer key. Solution by PrepWiser.