When a number or algebraic term is raised to the power zero, the result is 1, provided the base is not zero.

This is called the zero index or zero exponent rule.

Why the Rule Works

The rule follows naturally from the laws of indices. Consider dividing a power by itself:

But any non-zero quantity divided by itself equals 1, so $a^0 = 1$.

Applying the Rule

• Any non-zero base raised to the power zero simplifies to 1: $5^0 = 1$, $x^0 = 1$, $(-3)^0 = 1$.
• The base can be a number, a variable, or a whole expression: $(2x + 7)^0 = 1$.
• Coefficients in front of a zero-index term are kept: $4x^0 = 4 \times 1 = 4$.
• The rule applies only to the power zero, not to other powers: $x^0 + x^1 = 1 + x$.

Important Exceptions and Cautions

• The base must be non-zero. The expression $0^0$ is undefined.
• Be careful to identify the base correctly. In $2x^0$, only $x$ is raised to the power zero, so the result is $2 \times 1 = 2$. In $(2x)^0$, the whole product is raised to zero, so the result is 1.
• Negative bases still give 1: $(-8)^0 = 1$.

Combining with Other Index Laws

When simplifying expressions, apply the zero index rule alongside other index laws:

• Multiplication: $a^0 \times a^3 = 1 \times a^3 = a^3$.
• Division: $\frac{a^5}{a^5} = a^0 = 1$.
• Powers of powers: $(a^0)^4 = 1^4 = 1$.

Summary

• $a^0 = 1$ for any non-zero $a$.
• Identify the base carefully before simplifying.
• Keep any coefficients outside the zero-index term.
• Remember that $0^0$ is undefined.