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### 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

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.

that puzzles you, confuses you, or makes you curious.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents
High School Negative Numbers

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