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Math Help - Negative Exponents of i

  1. #1
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    Question Negative Exponents of i

    Hello all,
    I know that the imaginary number unit i can be raised to powers and get either -1, 1, i, or -i.

    However, I have come across a problem where I have to find i to negative exponents. How do I do this? Is there a rule or pattern like there is in positive exponents?

    Thanks!
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  2. #2
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    Quote Originally Posted by scotland4ever View Post
    Hello all,
    I know that the imaginary number unit i can be raised to powers and get either -1, 1, i, or -i.

    However, I have come across a problem where I have to find i to negative exponents. How do I do this? Is there a rule or pattern like there is in positive exponents?

    Thanks!
    Like for all other negative exponents:
    a^{-n} = \frac{1}{a^n}

    So, for example
    i^{-1} = \frac{1}{i} = -i (after you rationalize the denominator.)

    i^{-2} = \frac{1}{i^2} = -1

    i^{-3} = \frac{1}{i^3} = i

    i^{-4} = \frac{1}{i^4} = 1

    Notice the pattern?

    -Dan
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  3. #3
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    One of the most useful complex number properties to know is: z^{ - 1}  = \frac{{\overline z }}{{\left| z \right|^2 }}

    Examples: \left( {3 - 4i} \right)^{ - 1}  = \frac{{3 + 4i}}{{25}}

    \left( i \right)^{ - 7}  = \left( {i^7 } \right)^{ - 1}  = \left( { - i} \right)^{ - 1}  = \frac{{\overline {\left( { - i} \right)} }}{{\left| { - i} \right|^2 }} = i.
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  4. #4
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    Plato, what is the line over z and (-i)? What does it mean?

    Thank you!
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