Zero Exponent
In short: A zero exponent makes any nonzero base equal to 1, so a to the power 0 is 1 for every a that is not zero, whether the base is a whole number, a fraction or negative.
8th GradeNumbers and Operations8.EE.A.1
What is Zero Exponent?
A zero exponent sounds like it should give zero, but it gives 1: \(a^0 = 1\) whenever \(a \neq 0\). Two independent arguments agree. A quotient of identical powers, \(\frac{4^5}{4^5}\), equals 1, while the quotient rule gives \(4^{5-5} = 4^0\).
The pattern argument gives the same answer: halving down a list of powers of 2 yields \(2^3 = 8\), \(2^2 = 4\), \(2^1 = 2\), and the next entry must be \(2^0 = 1\). The expression \(0^0\) is left undefined at this level.
How to spot it
- An expression contains a power of zero, such as seven to the zero or a bracket raised to the zero.
- A quotient rule subtracts equal exponents and leaves a zero.
- A variable carries the exponent zero and the problem states that the variable is not zero.
- A negative sign sits outside a base that has the exponent zero, and you must decide what it applies to.
The method
- Confirm that the base is not zero; this rule does not apply to zero itself.
- Replace the whole power that has exponent 0 with the number 1.
- Handle any coefficient or minus sign outside the power separately, since it multiplies the 1.
- Finish the arithmetic with the remaining terms.
Zero Exponent: solved example
Problem
Evaluate \(7^0\), \((-12)^0\), \(-3^0\) and \(5x^0\) for \(x \neq 0\).
Solution
\(7^0 = 1\) and \((-12)^0 = 1\), since the whole bracket is raised to zero.
In \(-3^0\) the exponent touches only the 3, so \(-3^0 = -(3^0) = -1\). And \(5x^0 = 5 \cdot 1 = 5\). The values are \(1,\ 1,\ -1,\ 5\).
Try it
Question 1
Use the quotient rule to explain why \(\frac{9^4}{9^4}\) equals \(9^0\), and give the value.
Show the answer
The quotient rule gives \(9^{4-4} = 9^0\), and any number divided by itself is 1, so \(9^0 = 1\).
Question 2
Evaluate \((3x)^0 + 3x^0\) when \(x \neq 0\).
Show the answer
\((3x)^0 = 1\) and \(3x^0 = 3 \cdot 1 = 3\), so the sum is 4.
Zero Exponent vs. Negative Exponent
A zero exponent returns exactly 1 no matter which nonzero base is used. A negative exponent returns a reciprocal that depends on the base: \(2^{-3} = \frac{1}{8}\) but \(5^{-3} = \frac{1}{125}\). Zero marks the point in the pattern where powers pass from whole-number sizes into fractions.
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