Question

If $\infty$ is an imaginary cube root of unity, then $(1+\infty )(1+{\infty }^2)(1+{\infty }^4)(1+{\infty }^8)$ ........$100$ factors is equal to

• A. $1$
• B. $-1$
• C. $\infty$
• D. ${\infty }^2$

Answer 1

Answer: (A)

$1$

Given, $\infty$ is an imaginary cube root of unity.
$(1+\infty )(1+{\infty }^2)(1+{\infty }^4)(1+{\infty }^8)$ ........$100$ factors
$=(-{\infty }^2)(-\infty )(-{\infty }^2)(-\infty )(-{\infty }^2)(-\infty )$.....$100$ factors
$=\left({\infty }^3\right)\left({\infty }^3\right)\left({\infty }^3\right)$........$50$ times
$=1$
Hence, option A.