Exploring Zero and Negative Exponents

Property As the exponent of a base decreases by 1, the value of the power is divided by the base. Following this pattern past the exponent of 1 reveals the rules for zero and negative exponents: For every nonzero number $x$, $x^0 = 1$. For every nonzero number $x$, $x^{ n} = \frac{1}{x^n}$.

Examples Consider the pattern for powers of 2: $$2^3 = 8$$ $$2^2 = 4 \quad (\text{since } 8 \div 2 = 4)$$ $$2^1 = 2 \quad (\text{since } 4 \div 2 = 2)$$ $$2^0 = 1 \quad (\text{since } 2 \div 2 = 1)$$ $$2^{ 1} = \frac{1}{2} \quad (\text{since } 1 \div 2 = \frac{1}{2})$$ $$2^{ 2} = \frac{1}{4} \quad (\text{since } \frac{1}{2} \div 2 = \frac{1}{4})$$ Notice that $2^{ 2}$ is exactly the same as $\frac{1}{2^2}$. Simplify $\frac{x^{ 4}}{y^2}$: Apply the "flip" rule to the negative exponent to move it to the denominator, resulting in $\frac{1}{x^4 y^2}$.

Explanation Zero and negative exponents aren't magic; they are just the logical continuation of a mathematical pattern! Every time an exponent drops by one, you divide by the base. This proves why any non zero number to the power of zero is exactly 1. It also shows that a negative exponent is basically a "flip it" command: it tells you to take the reciprocal of the base and make the exponent positive.