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Negative Exponents and Scientific Notation

Having successfully defined 0 as an exponent, it seems reasonable (to a mathematitian) to define negative integers as exponents next.

As before, we look to the quotient rule as our guide. To decide how to define a negative exponent, look at a quotient in which the power on the bottom is bigger than the power on the top.

Now look at the same problem, but try applying the quotient rule to it instead:

Thus, it makes sense to say that

Definition (Negative Integers as Exponents)

For any real number a ( a 0) and any integer n,

Note that there is no negative sign on the right-hand side of the equation in the definition above.
This definition also tells us that (Why?)

The usual rules for powers hold with this definition.

Scientists often deal with very large (such as 93,000,000,000,000) and very small numbers (such as 0.00000000000113). To help deal with it, they have a special notation, called scientific notation which they use to write these numbers.

Definition (Scientific Notation)

Scientific notation for a number is an expression which looks like N Ã—10ⁿ, where 1 N <10 and n is any integer.

This means that any number written properly in scientific notation will have a single nonzero digit before the decimal place.