A 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.
When the denominator has more factors than the numerator, the leftover factors stay below the bar: 2325 = 122 = 2−2. The row shows the same pattern: each step to a smaller exponent divides by 2.
Formal definition and notation
For a real numbera ≠ 0 and a positive integern, a−n is defined as 1an. This is the only definition that keeps the product rule am · an = am + n true for all integers, because an · a−n = a0 = 1. It follows that 1a−n = an and ( ab )−n = ( ba )n for nonzero a, b.
A negative exponent means “reciprocal”, not “negative”: 5−2 = 125, a positive number.
For a base greater than 1, a negative exponent gives a value between 0 and 1: 4−1 = 0.25.
A fraction to a negative power flips: ( 23 )−2 = ( 32 )2 = 94.
A negative exponent in the denominator moves up: 1x−4 = x4.
Small numbers in scientific notation use negative powers of 10: 0.0007 = 7 × 10−4.
The pattern continues the zero exponent: 31 = 3, 30 = 1, 3−1 = 13.
Negative exponent rule
Negative exponent rule
a−n = 1an and 1a−n = an
a
the base, any nonzero number or expression
n
a positive number; −n is the negative exponent
Moving a factor across the fraction bar changes the sign of its exponent. Only the factor with the negative exponent moves; everything else stays put.
Try it: go below zero
Interactive
The explorer starts at 2−3. Step the exponent from 3 down to −4 and watch the value divide by the base each time; the readout shows each negative power as 1 over a product.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
A negative number is less than 0, like −8. A negative exponent gives a reciprocal that is still positive for a positive base: 2−3 = 18, not −8.
Negative Exponentvs.Zero exponent
An exponent of 0 gives 1; a negative exponent gives a fraction: 60 = 1 but 6−1 = 16.
Negative Exponentvs.Inverse function
In f−1(x) the −1 names the inverse function, not a reciprocal. A negative exponent on a number, like x−1 = 1x, does mean reciprocal.
Quick check: 3 questions
Question 1 of 3
What is 10−3?
Show answer and explanation
Answer: B.10−3 = 1103 = 11000 = 0.001. −1000 makes the value negative, −30 multiplies base by exponent, and 0.0001 has one zero too many.
Question 2 of 3
Simplify 1m−5.
Show answer and explanation
Answer: D. A factor with a negative exponent in the denominator moves to the numerator and its exponent becomes positive: 1m−5 = m5. 1m5 is the same as m−5, the reciprocal of the answer.
Question 3 of 3
Which expression equals 149?
Show answer and explanation
Answer: A.7−2 = 172 = 149. −72 = −49, 49−2 = 12401, and 712 is the square root of 7.
Negative Exponent: frequently asked questions
Does a negative exponent make a number negative?
No. A negative exponent makes a reciprocal. 3−2 = 19, which is positive. The result is negative only when the base itself is negative and the power is odd, such as (−2)−3 = −18. Think “flip”, not “minus”, whenever you see one.
How do you get rid of a negative exponent?
Move the factor to the other side of the fraction bar and make the exponent positive. A factor in the numerator such as x−4 becomes 1x4, and a factor in the denominator such as 1y−2 becomes y2. Factors with positive exponents stay where they are.
Why is a to the negative n equal to 1 over a to the n?
Because exponents must keep adding when you multiply powers. a3 · a−3 should be a3 + (−3) = a0 = 1, so a−3 has to be the number that multiplies a3 to give 1: its reciprocal, 1a3. The pattern 8, 4, 2, 1, one half for powers of 2 shows the same thing.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.EE.A.1
Cite this entry
Effortless Math Team. “Negative Exponent.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/negative-exponent/
Flashcard
TermNegative Exponenta−n = ?
DefinitionA negative exponent means take the reciprocal of the positive power: a−n = 1an. Example: 2−3 = 18, still positive.
Real 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 1Positive IntegerA positive integer is a whole number greater than zero: 1, 2, 3, 4 and so on. Positive integers have no fraction or decimal part and are never 0 or negative. They are also called counting numbers, and the set is written ℤ+. When a test says n is a positive integer, n cannot be 0, −3 or 2.5.Problem Solving · Grade 6, Grade 7, Algebra 1ReciprocalThe 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 7Scientific 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 1NumberA 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 2PowerA 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 7DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5Zero ExponentA 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.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 6ExponentAn 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 8NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7Inverse FunctionAn inverse function undoes another function: if f sends a to b, its inverse f−1 sends b back to a. For f(x) = 2x + 3, the inverse is f−1(x) = x − 32. Only one-to-one functions have inverses, and the graph of f−1 is the mirror image of f across y = x.Functions · Algebra 2, PrecalculusFractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5Rational ExponentA rational exponent is an exponent written as a fraction, and it stands for a root. The denominator is the root and the numerator is the power: a1n = n√a and amn = ( n√a )m. For example, 912 = 3 and 2723 = 32 = 9.Algebra · Algebra 1, Algebra 2Exponential DecayExponential decay is a decrease by the same percent in each equal time period, so the amount drops quickly at first and then more and more slowly. It is modeled by y = a(1 − r)t, where a is the starting amount and r is the decay rate. The amount approaches 0 but never reaches it.Functions · Algebra 1, Algebra 2Laws 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 2BaseA 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 8