Calculating i raised to arbitrary exponents | Precalculus | Khan Academy

Main ideas / concepts

• The powers of the imaginary unit (i) repeat in a cycle: [ i^1=i,\quad i^2=-1,\quad i^3=-i,\quad i^4=1 ]

• For higher powers, (i^n) cycles through: [ 1,\ i,\ -1,\ -i,\ 1,\ i,\ -1,\ -i,\ \dots ]

• This cyclic behavior allows computing very large exponents “quickly” by reducing the exponent modulo (4), rather than repeatedly multiplying.

• Key exponent rules used:
  • ((a^m)^n = a^{mn})
  • (a^m \cdot a^n = a^{m+n})

The video then applies these ideas to compute specific examples.

Method / procedure shown (detailed)

To compute (i^n) efficiently:

• Use the cycle of period 4:

  • (i^4 = 1)
  • Therefore, for integers (k \ge 0): [ i^{4k} = (i^4)^k = 1^k = 1 ]

• If the exponent is not a multiple of 4, write it as a multiple of 4 plus a remainder:

  • Let [ n = 4k + r,\quad \text{where } r \in {0,1,2,3}. ]

  • Then: [ i^n = i^{4k+r} = i^{4k}\cdot i^r = (i^4)^k \cdot i^r = 1 \cdot i^r = i^r ]

• Compute (i^r):

  • If (r=0), then (i^n = 1)
  • If (r=1), then (i^n = i)
  • If (r=2), then (i^n = -1)
  • If (r=3), then (i^n = -i)

Worked examples from the subtitles

• Compute (i^{100}):

  • (100) is a multiple of 4: (100=4\cdot 25) [ i^{100} = \left(i^4\right)^{25} = 1^{25} = 1 ]

• Compute (i^{501}):

  • (501 = 500 + 1), and (500=4\cdot 125) [ i^{501} = i^{500}\cdot i^1 = \left(i^{4}\right)^{125}\cdot i = 1\cdot i = i ]

  • General pattern stated:

    • if (n=4k), then (i^n=1)
    • if (n=4k+1), then (i^n=i)

• Compute (i^{7321}):

  • (7321 = 7320 + 1), and (7320) is divisible by 4 [ i^{7321} = i^{7320}\cdot i = 1\cdot i = i ]

• Compute (i^{99}):

  • The largest multiple of 4 less than 99 is 96, so (99 = 96 + 3) [ i^{99} = i^{96}\cdot i^3 = 1\cdot i^3 = i^3 = -i ]

  • Alternate derivation shown: [ i^3 = i^2\cdot i = (-1)\cdot i = -i ]

• Compute (i^{38}):

  • (38 = 36 + 2) and (36) is divisible by 4 [ i^{38} = i^{36}\cdot i^2 = 1\cdot (-1) = -1 ]

Speakers / sources featured

• No specific speaker name is provided in the subtitles.
• Source implied by the video title: Khan Academy (inferred from the title).