Zero, Negative, and Fractional Exponents

Zero exponent

Applying the quotient law to $\dfrac{a^m}{a^m}$ two different ways gives a natural definition for the zero exponent. On one hand, $\dfrac{a^m}{a^m}=1$ (any nonzero number divided by itself). On the other hand, the quotient law gives $\dfrac{a^m}{a^m}=a^{m-m}=a^0$. For both to agree:

Negative exponent

Similarly, applying the quotient law to $\dfrac{a^m}{a^n}$ when $n>m$ (so the direct subtraction $m-n$ is negative) and comparing with cancelling factors directly leads to:

A negative exponent means "take the reciprocal, then raise to the positive version of that exponent" — it never means the result itself becomes negative.

Fractional exponent (roots)

A fractional exponent represents a ROOT:

and more generally, combining this with power-of-a-power:

Both routes — taking the root first and then the power, or the power first and then the root — always give the same value, so whichever is arithmetically easier may be used.

The seven laws, side by side

| Law | Rule |

|---|---| …