# Negative Exponents of i

• Nov 14th 2007, 03:17 PM
scotland4ever
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!
• Nov 14th 2007, 03:21 PM
topsquark
Quote:

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:
$a^{-n} = \frac{1}{a^n}$

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
• Nov 14th 2007, 04:08 PM
Plato
One of the most useful complex number properties to know is: $z^{ - 1} = \frac{{\overline z }}{{\left| z \right|^2 }}$

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

$\left( i \right)^{ - 7} = \left( {i^7 } \right)^{ - 1} = \left( { - i} \right)^{ - 1} = \frac{{\overline {\left( { - i} \right)} }}{{\left| { - i} \right|^2 }} = i$.
• Nov 14th 2007, 04:32 PM
scotland4ever
Plato, what is the line over z and (-i)? What does it mean?

Thank you!