Question

Express the complex number in the form of a + ib: (2 – i)⁴
1. -1 – 2i
2. -7 + 24i
3. – 24i
4. -7 – 24i

Answer 1

Correct Answer - Option 4 : -7 – 24i

CONCEPT:

We know that for any integer k, \({i^{4k}} = 1,{\rm{\;}}{i^{4k + 1}} = i,{\rm{\;}}{i^{4k + 2}} = - 1,\;{i^{4k + 3}} = - i\)

Also (a + b)2 = a² + b² + 2ab

CALCULATION:

Given expression is (2 – i)⁴

∴ (2 – i)⁴ = [(2 – i)²]²

⇒ [2² + i² – 4i]² = [4 – 1 – 4i]² = [3 – 4i]²

⇒ [3² + (4i)² + 2 × 3 × (-4i)] = [9 – 16 – 24i]

⇒ z = -7 – 24i