Question

prove that (1+i)⁴×(1+1/i)⁴=16​​

Answer 1

Question:-

$Prove \: that \: (1 + i {)}^{4} \times \bigg(1 + \frac{1}{i} \bigg)^{4} = 16$

Solution:-

$(1 + i {)}^{4} \times \bigg(1 + \frac{1}{i} \bigg)^{4}$

$= \left[(1 + i) \bigg(1 + \frac{1}{i} \bigg) \right] ^{4}$

$= \left[(1 + i) \frac{(1 + i)}{i} \right]^{4}$

$= \left[ \frac{(1 + i)}{i} \right]^{4}$

$= \bigg( \frac{1 + 2i + {i}^{2} }{ {i}^{4} } \bigg)^{4}$

$= \frac{(1 + 2i - 1)^{4} }{1}$

$= \frac{(1 + 2i - 1)^{4} }{1} \: \: \: \: \red{ ....}\left[∵ {i}^{2} = - 1\right]$

$= 16 {i}^{4}$

$= 16 \: \: \: \: \: \: \: .... [∵ {i}^{4} = 1]$

LHS = RHS

Hence, proved

Answer:-

16.

Concept: Algebra of Complex Numbers