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Show that ii(12+i2)10+(12-i2)10 = 0 - Mathematics and Statistics

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Question

Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0

Sum
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Solution

`(1/sqrt(2) + "i"/sqrt(2))^2 = 1/2 + 2(1/sqrt(2)) ("i"/sqrt(2)) + "i"^2/2`

= `1/2 + "i" - 1/2` = i

∴ `(1/sqrt(2) + "i"/sqrt(2))^10 = [(1/sqrt(2) + "i"/sqrt(2))^2]^5 = i^5 = i^4 * i = 1*i = i     ...(i)`

Also, `(1/sqrt(2) - "i"/sqrt(2))^2 = 1/2 - 2(1/sqrt(2)) ("i"/sqrt(2)) + "i"^2/2`

= `1/2 - "i" - 1/2` = – i

∴ `(1/sqrt(2) - "i"/sqrt(2))^10 = [(1/sqrt(2) - "i"/sqrt(2))^2]^5 =(-i)^5 = i^4 * (-i) = 1*(-i) = -i      ...(ii)`

Adding (i) and (ii), we get

`(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = i – i = 0

Hence proved.

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Chapter 1: Complex Numbers - Miscellaneous Exercise 1.2 [Page 22]

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Balbharati Mathematics and Statistics 2 (Arts and Science) [English] Standard 11 Maharashtra State Board
Chapter 1 Complex Numbers
Miscellaneous Exercise 1.2 | Q II.10 | Page 22

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