Expert: Steve Holleran - 8/19/2008

Question
just heard about "taking a number to a negative power"  Can you explain how to do this?  Obviously 9.250000 squared is 85.5625and 9.25 to the 5th power is 67,718.708.  But what is 9.25 to the negative 5th power and more importantly- how do I figure out this answer?  Any help is much appreciated.  Thanks!!!

Answer
Hi Bernard,

Raising a number to a negative power involves making it a fraction.

Check out this pattern, for example:

   4^3 = 64  ;  4^2 = 16 (64/4) ;  4^1 = 4 (16/4)

notice that as the power goes down, you just keep dividing by 4.  To continue then, it follows that 4^0 = 1 (which it does),so then extending to negative exponents, you see that

4^-1 = 1/4 = 0.25  ;  4^-2 = 1/16  ;  4^-3 = 1/64 ; ...

A quick way to get these without using patterns is to define

                  4^-n = 1/ 4^n

So, 4^-4 = 1/ 4^4 = 1/256 .  Likewise, 5^-2 = 1/5^2 = 1/25 ;

3^-4 = 1/3^4 = 1/81 and so on.

In the example you ask about then, 9.25^-5 = 1/9.25^5 = 1/67718.7 or

0.00001476696

You can now do any negative power you want; just use the definition

                      x^-n = 1/x^n

(It also applies to variables:  y^-3 = 1/y^3)

Steve