The Zero Exponent Rule states that for any nonzero number (a), raising it to the power of 0 gives:

$$a^0 = 1 \quad \text{(provided } a \neq 0\text{)}$$

Why Is This True?

This rule is designed to be consistent with the other laws of exponents. Consider the quotient rule for exponents, which says:

$$\frac{a^m}{a^n} = a^{m-n} \quad \text{(for } a \neq 0\text{)}$$

If we let (m = n), then:

$$\frac{a^n}{a^n} = a^{n-n} = a^0$$

But we also know that:

$$\frac{a^n}{a^n} = 1 \quad \text{(since any nonzero number divided by itself equals 1)}$$

So, we conclude:

$$a^0 = 1$$

Examples

• ( 5^0 = 1 )
• ( (-3)^0 = 1 )
• ( \left(\frac{1}{2}\right)^0 = 1 )

Important Note

• The rule applies only to nonzero numbers. The expression (0^0) is considered indeterminate or undefined in most contexts because it leads to contradictory interpretations in different mathematical scenarios.

Summary

The Zero Exponent Rule is a convenient way to extend the laws of exponents consistently. By defining (a^0 = 1) for any nonzero (a), we maintain the validity of other exponent rules such as the quotient rule.

Latex

a^0 = 1 \quad \text{(provided } a \neq 0\text{)}