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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

The value of `i^n` for various values of n is discussed.



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What is the value of `i`?

  • `-1`
  • `sqrt(-1)`
  • `sqrt(-1)`

The answer is '`sqrt(-1)`'

What is the value of `i^2`?

  • `-1`
  • `-1`
  • `sqrt(-1)`

The answer is '`-1`'

It is defined that `i=sqrt(-1)`. Substitute that in the following
`i^2`
`quad quad = (sqrt(-1))^2`
`quad quad = (-1^(1/2))^2`
`quad quad = -1^(2/2)`
`quad quad = -1`

What is the value of `i^3`?

  • `-1`
  • `i`
  • `-i`
  • `-i`

The answer is '`-i`'

`i^3`
`quad quad = i^(2+1)`
`quad quad = i^2 xx i^1`
`quad quad = -1 xx i`
`quad quad = -i`

What is the value of `i^4`?

  • `1`
  • `1`
  • `-1`

The answer is '`1`'

`i^4`
`quad quad = i^(2+2)`
`quad quad = (i)^2 xx i^2`
`quad quad = -1 xx -1`
`quad quad = 1`

What is the value of `i^0`?

  • `1`
  • `1`
  • `-1`
  • `i`

The answer is '`1`'

`i^0`
`quad quad = i^(1-1)`
`quad quad = i/i`
`quad quad = 1`

What is the value of `i^n`, if `n=4p+2q+r`?

  • `i`
  • `(-1)^q xx i^r`
  • `(-1)^q xx i^r`
  • `i^r`

The answer is '`(-1)^q xx i^r`'

`i^n`
`quad quad = i^(4p+2q+r)`
`quad quad = i^(4p) xx i^(2q) xx i^r`
`quad quad = (i^4)^p) xx (i^2)^q) xx i^r`
`quad quad = (-1)^q xx i^r`

To calculate `i^n`, use `i^4=1` and `i^2=-1`.

Power of i: To calculate `i^n`, express `n` in the form `n=4p+2q+r`, where `p,q, r in NN` natural numbers. Then
`i^n = (-1)^q xx i^r`.

Solved Exercise Problem:

What is the value of `i^27`?

  • `1`
  • `-i`
  • `-i`
  • `i`

The answer is '`-i`'

Solved Exercise Problem:

What is `i^20`?

  • `1`
  • `1`
  • `-1`
  • `i`
  • `-i`

The answer is '`1`'

Solved Exercise Problem:

What is `i^7`?

  • `1`
  • `-1`
  • `i`
  • `-i`
  • `-i`

The answer is '`-i`'

                            
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