Question

The value of ${\left(1+i\right)}^8+{\left(1-i\right)}^8$ is

• A. $16$
• B. $-16$
• C. $32$
• D. $-32$

Answer 1

The correct option is C

$32$

Explanation for the correct answer:

Simplification of expression:

${\left(1+i\right)}^8={\left({\left(1+i\right)}^4\right)}^2$

Similarly,

${\left(1-i\right)}^8={\left({\left(1-i\right)}^4\right)}^2$

We know,

${\left(1+i\right)}^4={\left({\left(1+i\right)}^2\right)}^2$ $\left(\because {\left(a+b\right)}^2=a^2+2\mathrm{ab}+b^2\right)$

$={\left(1+2i-1\right)}^2$ $\left(\because i^2=-1\right)$

$={\left(2i\right)}^2$

$\Rightarrow$${\left(1+i\right)}^4$ $=-4$

$\Rightarrow$ ${\left(1+i\right)}^8=16$

Similarly,

$\Rightarrow$${\left(1-i\right)}^4={(-2i)}^2$ $\left(\because {\left(a-b\right)}^2=a^2-2\mathrm{ab}+b^2\right)$

$=-4$

$\Rightarrow$ ${\left(1-i\right)}^8=16$

$\therefore$ ${\left(1+i\right)}^8+{\left(1-i\right)}^8=16+16$

$=32$

Hence, the value of ${\left(1+i\right)}^8+{\left(1-i\right)}^8$ is $32$.

Hence, option(C) is the correct answer.