Are Negative Exponents Like Other Exponents?

Date: 08/21/2002 at 09:09:18
From: Stephanie
Subject: Are positive exponents like other exponents?

I know how to do positive and negative exponents. I know that anything 
to the power of zero is one. I don't know much about decimal 
exponents, except that n^0.5 is the square root of n. What seems 
strange to me is that positive exponents are done very differently 
from negative exponents, or decimal exponents, or zero as the 
exponent. Is there a general rule for doing all exponents, or does a 
negative exponent have nothing in common with positive exponents?  
They are all exponents, but they are done differently. Is there a way 
to do all exponents that is the same for a positive or negative or 
decimal exponent?  Are these exponents the same at all?

Stephanie

Date: 08/21/2002 at 10:44:01
From: Doctor Rick
Subject: Re: Are positive exponents like other exponents?

Hi, Stephanie.

All exponents follow the same rules. For instance, the product of the 
same number to two powers is the number to the sum of the powers:

  2^3 * 2^5 = 2^(3+5)

Also, if you raise a number to a power, then raise the result to 
another power, it's the same as raising the number to the product of 
the powers:

  (2^3)^5 = 2^(3*5)

A third important rule is that the product of two numbers to the same 
power is the product of the two numbers, raised to the power:

  2^3 * 5^3 = (2*5)^3

This last one is a "distributive property of powers over 
multiplication," much like the distributive property of multiplication 
over addition:

  2*3 + 5*3 = (2+5)*3

ALL exponents, whether positive, negative, integer or fractional, obey 
these same rules. We can use these rules to understand the meaning of 
negative or fractional exponents, based on our first understanding of 
whole-number exponents. For instance:

  (3^2)^(1/2) = 3^(2*1/2)
  9^(1/2) = 3^1
  9^(1/2) = 3

The multiplication property of exponents thus shows us that the 1/2 
power is the same as a square root. In the same way,

  (5^3)^(1/3) = 5^(3*1/3)
  125^(1/3) = 5^1
  125^(1/3) = 5

so we see that the 1/3 power of a number is its cube root - the number 
whose cube (third power) is the number.

With negative exponents, we find that

  2^3 * 2^-3 = 2^(3 + -3)
  2^3 * 2^-3 = 2^0
  2^3 * 2^-3 = 1
  2^-3 = 1/(2^3)

I got the last step by dividing each side of the equation by 2^3. The 
addition property of exponents thus shows us that a number to a 
negative power is the reciprocal of the number to the positive power.

Though they all follow the same rules for combining them, computing 
them is a very different matter. Raising a number to a whole-number 
power is just a matter of doing multiplications (one fewer than the 
power), but raising a number to a negative power requires a division 
as well. Raising a number to a fractional power, even the simplest 
(1/2), takes a lot of work!

But this shouldn't be so surprising. Remember when you first learned 
about negative numbers? Adding a negative number was different from 
adding a positive number: in fact, it was subtraction. And adding 
fractions was pretty complicated; I'll bet it took some time for you 
to get the hang of it. Still, positive numbers, negative numbers, 
and fractions follow the same rules: the distributive property of 
multiplication over addition, which I mentioned earlier, is true 
whether the numbers are positive, negative, integer, or fractional. 
It's just the same with exponents.

Does this answer your questions? I'd be glad to talk about anything 
that puzzles you, confuses you, or makes you curious.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/