Advertisement Remove all ads

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 \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].

Advertisement Remove all ads

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?
Advertisement Remove all ads

APPEARS IN

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

Video TutorialsVIEW ALL [1]

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×