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Graph of y = (-n)^x

Date: 01/17/2005 at 23:19:51
From: Alexander
Subject: A negative number to the power of x

I am curious as to what the graph of y = (-n)^(x) would look like, 
such as y = (-2)^x.  My graphing calculator will not show the graph as 
anything, but has many real values in a table of values.

It is confusing becuase a negative number to the power of let's say 
0.6 (3/5, so odd root) is a real number while a negative number to the
power of 0.7 (7/10, so even root) would be an imaginary number.  This
would mean there are many random points that are both real and
imaginary on a graph of this sort.

Is it possible to construct a graph of y = (-n)^x becuase of this 
interesting coincidence that happens with odd and even roots of the 
negative numbers?

Date: 01/18/2005 at 09:48:45
From: Doctor Vogler
Subject: Re: A negative number to the power of x

Hi Alexander,

Thanks for writing to Dr. Math.  It depends partly on what you mean by
"(-n)^x".  You see, generally exponents are either defined 
algebraically for rational exponents or continuously (through 
calculus) for positive bases.  If you have a negative base and want
all exponents (including irrationals) then things get ugly.  See also

  Base of an Exponential Function

Notice that in the case of all irrational and many rational exponents,
the only possible value for y is complex.  When graphing, you 
generally don't graph complex solutions.  (Did you plot (i+1, 2i) on
your graph of y = x^2?  I didn't think so.)  So your graph will
essentially be dotted lines where the rational values take you.  If
you plotted those values where x is rational, then you would have a
dotted line along

  y = n^x

(since y = n^x when x has an even numerator and odd denominator) and a
dotted line along

  y = -(n^x)

(since -y = n^x when x has an odd numerator and odd denominator).

If you want to consider what are the complex values of

  y = (-n)^x,

then you have a completely different beast, since there are more than
one possible values for y unless x is an integer, and there are
infinitely many such values unless x is rational.  See also

  Complex Powers

If you have any questions about this or need more help, please write
back, and I will try to offer further suggestions.

- Doctor Vogler, The Math Forum
Associated Topics:
High School Exponents
High School Imaginary/Complex Numbers
High School Number Theory

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