# For a positive integer n, find the value of ( 1 − i ) n ( 1 − 1 i ) n . - Mathematics

For a positive integer n, find the value of $(1 - i )^n \left( 1 - \frac{1}{i} \right)^n$.

#### Solution

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

$= \left( 1 - i \right)^n \left( 1 - i^3 \right)^n$

$= \left( 1 - i \right)^n \left( 1 + i \right)^n [ \because i^3 = - i]$

$= \left[ (1 - i)(1 + i) \right]^n$

$= (1 - i^2 )^n$

$= 2^n [ \because i^2 = - 1]$

Thus, the value of

$(1 - i )^n \left( 1 - \frac{1}{i} \right)^n$ is 2n.

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.2 | Q 17 | Page 32