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

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Sum

For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`

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Solution

We have `(1 - i)^n (1 - 1/i)^"n"`

= `[(1 - i)(1 - 1/i)]^n`

= `[(1 - i) (1 - 1/i xx i/i)]^n`

= `[(1 - i)(1 - i/i^2)]^n`

= `[(1 - i)(1 + i)]^n`  .....`[because i^2 = -1]`

= `[1 - i^2]^n`

= `[1 + 1]^"n"`

= 2n

Hence, `(1 - i)^n (1 - 1/i)^n` = 2n.

Concept: Algebraic Operations of Complex Numbers
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Chapter 5: Complex Numbers and Quadratic Equations - Exercise [Page 91]

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NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Exercise | Q 1 | Page 91

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