# Thread: Negative Exponents of i

1. ## 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!

2. Originally Posted by scotland4ever
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:
$\displaystyle a^{-n} = \frac{1}{a^n}$

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

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

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

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

Notice the pattern?

-Dan

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

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

$\displaystyle \left( i \right)^{ - 7} = \left( {i^7 } \right)^{ - 1} = \left( { - i} \right)^{ - 1} = \frac{{\overline {\left( { - i} \right)} }}{{\left| { - i} \right|^2 }} = i$.

4. Plato, what is the line over z and (-i)? What does it mean?

Thank you!