A zero exponent is an exponent of 0, and any nonzero number raised to the power 0 equals 1. So 70 = 1, (−3)0 = 1 and x0 = 1 for x ≠ 0. The rule keeps the quotient rule working: 5353 = 50, and a number divided by itself is 1.
In Spanish
exponente cero
Also called
zero exponent rule, zero power rule, power of zero, zero exponent property
Going down the ladder, each smaller exponent divides the value by the base, so 20 = 2 ÷ 2 = 1. Dividing a power by itself gives the same answer: 50 = 1.
Formal definition and notation
For every real numbera ≠ 0, a0 = 1. This is a definition chosen so that the quotient rule aman = am − n still holds when m = n: the left side is amam = 1 and the right side is a0. The expression 00 is an indeterminate form and is left undefined in algebra courses.
Symbol
Read it as
Meaning
a0
a to the zero power
equals 1 whenever a ≠ 0
5x0
5 times x to the zero
only x is raised to 0, so it equals 5 · 1 = 5
(5x)0
the quantity 5 x, to the zero
the whole product is raised to 0, so it equals 1
Key facts
The value of a0 is 1, not 0 and not a. Zero factors multiply to 1, the same way an empty sum is 0.
The sign or size of the base does not matter: 10000 = 1, ( 23 )0 = 1, (−8)0 = 1.
Watch what the exponent touches: 3x0 = 3 but (3x)0 = 1.
00 is undefined in algebra, because the rules for powers of 0 (always 0) and for zero exponents (always 1) disagree there.
The rule sits between positive and negative exponents: 21 = 2, 20 = 1, 2−1 = 12.
Zero exponent rule
Zero exponent rule
a0 = 1 (a ≠ 0)
a
the base: any nonzero number, variable or expression
0
the exponent: zero factors of the base
An exponent applies only to what sits directly under it. In −40 the base is 4, so −40 = −1; in (−4)0 the base is −4, so the value is 1.
Try it: step the exponent down to 0
Interactive
The base starts at 6 and the exponent at 3. Lower the exponent one step at a time and watch the value divide by 6 each time until it reaches 1 at exponent 0. Then change the base: the power 0 still gives 1.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
In 6y0 only y has the exponent 0, so 6y0 = 6 · 1 = 6.
In (6y)0 the whole product has the exponent 0, so (6y)0 = 1.
Subtract: 6 − 1 = 5.
Answer: The expression simplifies to 5.
Example 3: Zero exponent from the quotient rule
Test-style
If 32k38 = 1, what is the value of k?
Quotient rule: 32k38 = 32k − 8.
A power of 3 equals 1 only when its exponent is 0, so 2k − 8 = 0.
Solve: 2k = 8, so k = 4.
Check: 3838 = 1.
Answer: k = 4
Common mistakes
✗ Mistake80 = 0, because “zero times” gives nothing.
✓ Instead80 = 1.
An exponent counts factors, not a product with 0. With no factors of 8 left, what remains is the starting value of any product, which is 1.
✗ Mistake80 = 8, since the base “stays the same”.
✓ Instead81 = 8, but 80 = 1.
An exponent of 1 leaves the base unchanged. An exponent of 0 is one more division by 8: 8 ÷ 8 = 1.
✗ Mistake−50 = 1.
✓ Instead−50 = −(50) = −1, while (−5)0 = 1.
Without parentheses the exponent applies only to 5. The negative sign is applied after the power.
Not to be confused with
Zero Exponentvs.Zero Property of Multiplication
Multiplying by 0 gives 0: 8 × 0 = 0. Raising to the power 0 gives 1: 80 = 1. Same digit, different operation.
Zero Exponentvs.Negative exponent
A zero exponent gives 1. A negative exponent gives a reciprocal: 40 = 1, but 4−1 = 14.
Quick check: 3 questions
Question 1 of 3
What is 150?
Show answer and explanation
Answer: B. Any nonzero base to the power 0 is 1. 0 confuses the exponent with multiplying by 0, 15 is 151, and 115 is 15−1.
Question 2 of 3
Simplify 4a0 for a ≠ 0.
Show answer and explanation
Answer: D. Only a is raised to the power 0, so 4a0 = 4 · 1 = 4. Answering 1 treats it as (4a)0; 4a ignores the exponent.
Question 3 of 3
What is −20 + (−2)0?
Show answer and explanation
Answer: A.−20 = −1 because the exponent applies only to 2, and (−2)0 = 1. The sum is −1 + 1 = 0. Choice 2 treats both as 1.
Zero Exponent: frequently asked questions
Why does anything to the power of 0 equal 1?
Because each step down in the exponent divides by the base. 32 = 9, 31 = 3, and one more step gives 3 ÷ 3 = 1. The quotient rule says the same thing: 3434 = 30, and any nonzero number divided by itself is 1. Defining a0 = 1 keeps every exponent rule consistent.
Is 0 to the power of 0 equal to 1?
In algebra classes, 00 is left undefined. Powers of 0 such as 03 are 0, while zero powers such as 50 are 1, so the two patterns disagree at 00. Some areas of math, such as counting and power series, agree to use 00 = 1 as a convenience, but on a test, the answer is undefined.
Does the zero exponent rule work for negative numbers and fractions?
Yes. Any nonzero base works: (−7)0 = 1 and ( 34 )0 = 1. Just check where the parentheses are, because they decide what the base is. −70 means the opposite of 70, which is −1, because the exponent touches only the 7.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.EE.A.1
Cite this entry
Effortless Math Team. “Zero Exponent.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/zero-exponent/
Flashcard
TermZero Exponenta0 = ?
DefinitionAny nonzero number to the power 0 equals 1, because dividing a power by itself gives 1. Example: 120 = 1; 00 is undefined.
ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7Negative ExponentA negative exponent tells you to take the reciprocal of the base raised to the matching positive exponent: a−n = 1an for a ≠ 0. So 2−3 = 18 and 10−2 = 0.01. A negative exponent makes a number smaller in size, but it never makes the value negative.Algebra · Grade 8, Algebra 1EvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5ReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6, Grade 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SeriesA series is the sum of the terms of a sequence. The sequence 3, 7, 11, 15 is a list; the series 3 + 7 + 11 + 15 = 36 is what you get by adding it. A finite series stops after n terms, written Sn. An infinite series has a sum only if its partial sums approach a fixed number.Functions · Algebra 2, Precalculus, CalculusBaseA base is the number or expression that is repeatedly multiplied when using an exponent. In 45 the base is 4 and the exponent is 5, so 45 = 4 × 4 × 4 × 4 × 4 = 1024. The base is the big number written on the line; the exponent is the small raised number.Algebra · Grade 6, Grade 7, Grade 8Scientific NotationScientific notation writes a number as a value from 1 to less than 10 times a power of 10, such as 5.4 × 103 for 5,400. The exponent records how far the decimal point moved: positive for large numbers and negative for numbers between 0 and 1, as in 0.0062 = 6.2 × 10−3.Number System · Grade 8, Algebra 1Exponential FunctionAn exponential function is a function in which the input is the exponent: f(x) = a · bx, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Functions · Algebra 1, Algebra 2, PrecalculusLaws of ExponentsThe laws of exponents are the rules for simplifying powers that share a base or an exponent. Multiply powers of the same base by adding exponents, divide by subtracting them, and raise a power to a power by multiplying them: x2 · x5 = x7, x6x2 = x4, (x3)2 = x6.Algebra · Grade 8, Algebra 1, Algebra 2MonomialA monomial is a single term made of a number, a variable, or a product of numbers and variables with whole-number exponents, such as 8, x, or −7x2y. It has no addition or subtraction, and no variable in a denominator. Its degree is the sum of the exponents on its variables.Algebra · Grade 8, Algebra 1ZeroZero (0) is the number that represents none or no quantity. It is neither positive nor negative and sits in the middle of the number line, between the negative and positive numbers. Adding zero to a number does not change it, as in 9 + 0 = 9, so zero is called the additive identity.Number System · Grade 3, Grade 4, Grade 5