Zero Exponent Rule
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{)}