Zero and Negative Exponents
Derive that any nonzero base raised to zero equals 1 (using the quotient rule) and that negative exponents represent reciprocals. Simplify expressions containing zero and negative exponents without a calculator.
A lesson of Algebra 1, in Exponents & Exponential Functions, about 30 minutes.
By the end of this lesson, you will be able to say why $x^0 = 1$ rather than accepting it, convert between negative exponents and reciprocals, move factors across a fraction bar, and simplify an expression to positive exponents only.
Why Anything to the Zero Is 1
Counting cannot answer this. "Multiply $x$ by itself zero times" describes nothing. So the answer comes from the quotient rule instead.
$$\frac{x^3}{x^3} = x^{3-3} = x^0$$
But that fraction is a number divided by itself, which is $1$. Two correct routes, one expression:
$$x^0 = 1$$
It is forced, not decreed. Any other value would make the quotient rule false, and the rule is what everything else depends on. Mathematics chose the definition that keeps the existing structure intact, which is how every extension in this lesson works.
The pattern of dividing by the base makes it feel inevitable:
$$2^3 = 8 \qquad 2^2 = 4 \qquad 2^1 = 2 \qquad 2^0 = ?$$
Each step halves. The next number after $2$ is $1$.
$0^0$ is the one exception, because $\frac{0^3}{0^3}$ is dividing by zero and the derivation collapses. It is left undefined in this context.
The exponent applies only to its base. In $3x^0$, the $x^0$ is $1$, so the whole thing is $3$, not $1$.
Negative Exponents Are Reciprocals
Continue the halving past zero:
$$2^1 = 2 \qquad 2^0 = 1 \qquad 2^{-1} = \tfrac{1}{2} \qquad 2^{-2} = \tfrac{1}{4}$$
$$x^{-n} = \frac{1}{x^n}$$
The quotient rule forces this one too:
$$\frac{x^2}{x^5} = x^{2-5} = x^{-3}$$
And canceling the same fraction by hand leaves $\frac{1}{x^3}$. Both are correct, so they are equal.
A negative exponent is not a negative number.
$$2^{-3} = \frac{1}{8} \qquad \text{not } -8$$
The sign describes which side of the fraction bar the factor belongs on, and it never makes the value negative. That mistake is the single most common one in the topic, and the halving pattern above is the fastest way to see why it cannot be right: the numbers were shrinking toward zero, never turning negative.
Sort It: What Does It Equal
None of these is negative. Every answer is a positive number, which is the point of the sort.