Question

Prove that
$(1 - i) ^{4} = - 4$
​

Answer 1

$\mathfrak{\underline{\underline{\large{Answer:-}}}}$

$\bf{We\: have}$

$\tt{(1 - i) ^{4} = (1 - i {)}^{2} \times (1 - i {)}^{2} }$

$\tt{= (1 + {i}^{2} - 2i) \times (1 + i ^{2} - 2i)}$

$\tt{( - 2i)( - 2i) = 4 {i}^{2} }$

$\tt{ = 4 \times ( - 1) = - 4}$

Answer 2

$\huge\pink{Solution}$

LHS = ( 1-i )⁴ =>

We know that (a)⁴ can be written as (a)²(a)² . So ,

( 1-i )²(1-i)²

Applying (a-b)² identity that is a² +b² -2ab ,So

(1 +i² -2i )(1+i²-2i)

we know that ,

i = √-1 so , i² = (√-1)(√-1)

i² = -1

( +1-1-2i ) (+1-1-2i )

(-2i )(-2i )

4 i² as i² is -1 , hence

-4 = RHS

Hence proved.