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Negative Exponents

Objective Learn the notation of negative and zero exponents.

The concept of exponentiation as repeated multiplication cannot be extended to negative exponents. Therefore, this lesson may be conceptually difficult.


Negative Numbers and Zero in the Exponent

First, let's review the rule for dividing powers that have the same base.

Key Idea

When we divide a power of a by another smaller power of a, the result is a power of a, where the exponent is the difference of the exponents of the two dividends. In symbols,

Only exponentiation with positive integers has been defined. We have not discussed exponentiation with zero or with negative numbers. In the formula

what happens if we no longer insist that b be greater than c? What definition would this suggest to us for zero and negative exponents?

If we applied the formula for b = c, we would get

On the other hand, . So, the formula suggests that a 0 = 1.

Key Idea

For any nonzero number a, a 0 = 1.

What does the equation shown above suggest about negative numbers?

Let's consider a -1. According to the formula, we have

On the other hand,

So the formula suggests that . In general, we define a -n as .

Key Idea

For any nonzero number a and any integer n , .

You may be confused by the way the formula is used to generate a definition for negative powers. It is used as a guide to suggest what a definition should be, but that once we make the definition, we can work with these exponents in exactly the same way as we did with positive exponents. With this definition, these exponents satisfy all the laws of exponents previously studied.

The following formula shows what happens when b is less than c.


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