Key takeaway: A zero exponent makes any nonzero base equal to 1, so \(a^{0} = 1\) whenever \(a \neq 0\).
Grade 88.EE.A.1Numbers and Operations
Definition of Zero Exponent
The zero exponent rule says that any nonzero number raised to the power 0 equals 1. That looks strange at first, since 0 might suggest “nothing,” but an exponent counts how many times the base is multiplied, and multiplying zero times leaves the neutral starting value, 1.
Two quick arguments confirm that a zero exponent gives 1. First, divide equal powers: \(\frac{3^4}{3^4} = 1\), while the quotient rule gives \(3^{4-4} = 3^0\). So \(3^0\) must be 1. Second, follow the pattern 81, 27, 9, 3, where each step divides by 3: the next value is 1. The base must not be zero, because \(0^0\) is left undefined at this level.
Formula and notation
The rule is written as
\[a^{0} = 1, \qquad a \neq 0\]
The exponent attaches only to what it touches: \((5x)^{0} = 1\), but \(5x^{0} = 5 \cdot 1 = 5\).
Key facts
- Whatever the nonzero base is, large, small, fractional or negative, the result is 1.
- Parentheses decide what the exponent applies to, so check them carefully.
- A zero exponent never makes the answer 0; reading it that way is the most common slip.
- Variables work the same way: \(n^{0} = 1\) as long as n is not 0.
Zero Exponent example
Assuming \(x \neq 0\), simplify \(9 + 4x^{0} – (2x)^{0}\).
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The exponent in \(4x^{0}\) touches only \(x\), so \(x^{0} = 1\) and the term is \(4 \cdot 1 = 4\).
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In \((2x)^{0}\) the exponent covers the whole bracket, so it equals 1.
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Combine: \(9 + 4 – 1 = 12\).
The expression equals 12.
Common mistake
Watch out
Ravi decides that \(6^{0} = 0\), since zero times anything is zero.
Why it is wrong
An exponent is not a multiplier here. \(6^{0}\) means the empty product, which is 1, just as adding zero numbers gives 0 and multiplying zero numbers gives 1. Check with division: \(\frac{6^2}{6^2} = \frac{36}{36} = 1\), and the quotient rule says that same quotient is \(6^{2-2} = 6^{0}\).
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Numbers and Operations