Textbook
For a positive integer $k$, the power $a^k$ is the product of $k$ factors equal to $a$. For $a\ne0$, the definitions are $a^0=1$ and $a^{-k}=1/a^k$. A negative exponent takes a reciprocal; it does not negate the base. The expression $0^0$ needs a separate convention.
For a positive base, fractional exponents satisfy $a^{1/k}=\sqrt[k]{a}$. Other bases require their own domain checks. Parentheses determine the base: $(-2)^2=4$, whereas $-2^2=-(2^2)=-4$. A superscript can also be a label, as in $x^{(2)}$; the author must specify that role.
For integer exponents and a nonzero base, $a^p a^q=a^{p+q}$. With positive exponents, this follows by joining the chains of factors; the definitions for zero and negative exponents preserve the same rule. In particular, $a^k a^{-k}=a^0=1$, explaining why the reciprocal is required. Excluding zero matters here because zero has no multiplicative inverse.
Definitions and further reading: Basic Analysis I.
In plain language
A power packages repeated equal factors: $2^3$ uses three factors of two, unlike $2\cdot3$. A nonzero base raised to zero gives the multiplicative identity, while a negative exponent gives the reciprocal of the corresponding positive power. The minus changes the direction of scaling rather than the number’s mood. Parentheses delimit the entire base: a minus inside and a minus before the completed result can send the calculation to different addresses.
Example
The example uses base 2 with exponents 3, 0, and −3. The positive power is expanded first, followed by its reciprocal. The last operation would be forbidden for a zero base.
Exponentiation: Repeated Multiplication
Three factors
Three factors equal to 2 give 8.
Step 1 of 3: Three factors
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Three factors
Three factors equal to 2 give 8.
- State
- 2current2current2current8current
Zero exponent
A nonzero base raised to zero gives 1.
- State
- 2current0current1current
Reciprocal
For exponent −3, the result is the reciprocal of the previously computed 8.
- State
- 8current1/8current
Multiplication checks the transition from 8 to $1/8$ by returning the result to 1. The zero power also equals 1, but does not follow a rule of “base times exponent.” That rule would give zero and already contradict power laws. The base stays 2 in all three states; the exponent changes, along with the corresponding treatment of repeated factors.
Related concepts: Roots.