A 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.
In Spanish
exponente racional
Also called
fractional exponent, fractional power, exponent that is a fraction
In 823 the 3 says “cube root” and the 2 says “square”. Either order gives 4, but the root-first route uses smaller numbers.
Formal definition and notation
For a real numbera and integers m and n ≥ 2, with the fraction mn in lowest terms, amn = ( n√a )m = n√am whenever n√a is a real number. If n is even, a must be nonnegative. This definition is chosen so the power-of-a-power law holds: ( a1n )n = a1 = a.
A rational number is a value such as 23. A rational exponent is a fraction used as an exponent: in 823 the exponent is rational, and the result, 4, happens to be rational too, but 212 is irrational.
A square root is the special case with exponent one half: √49 = 4912 = 7. Rational exponents also cover cube roots, fourth roots and powers of roots.
Quick check: 3 questions
Question 1 of 3
What is 8114?
Show answer and explanation
Answer: C. The denominator 4 means fourth root: 34 = 81, so the answer is 3. 20.25 is 81 divided by 4, 9 is the square root, and one third would be 81−14.
Question 2 of 3
Which is equal to 5√x2?
Show answer and explanation
Answer: A. Power over root: power 2, root 5, so x25. x52 flips the fraction, x10 multiplies, and x−3 subtracts.
Question 3 of 3
Evaluate 432.
Show answer and explanation
Answer: B. Root first: √4 = 2, then 23 = 8. 6 multiplies 4 by three halves, 64 is 43 without the root, and one eighth is 4−32.
Rational Exponent: frequently asked questions
What does an exponent of 1/2 mean?
An exponent of one half means square root: x12 = √x. It has to, because the power-of-a-power law gives ( x12 )2 = x1 = x, and the number whose square is x is the square root. So 3612 = 6, 10012 = 10 and 212 ≈ 1.414.
How do you turn a radical into a rational exponent?
Put the power inside the radical over the index of the root. 3√x2 becomes x23, and √x5 becomes x52, since a square root has index 2. To go back, the denominator becomes the root index and the numerator becomes the power.
Can the base be negative with a fractional exponent?
Sometimes. When the root is odd, a negative base works: (−27)13 = −3. When the root is even, it does not give a real number: (−4)12 has no real value, because no real number squares to −4. That is why many rules state a > 0 or a ≥ 0.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSN-RN.A.1, CCSS HSN-RN.A.2
Cite this entry
Effortless Math Team. “Rational Exponent.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/rational-exponent/
Flashcard
TermRational Exponentamn: which number is the root?
DefinitionA fractional exponent means a root: the denominator is the root, the numerator the power. Example: 2723 = 32 = 9.
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 1NonnegativeNonnegative means greater than or equal to zero: such a number is either 0 or positive, so x ≥ 0. The nonnegative integers are 0, 1, 2, 3 and so on. It is not the same as positive, because 0 is nonnegative but not positive. Distances, absolute values and principal square roots are always nonnegative.Problem Solving · Grade 6, Grade 7, Algebra 1Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1Cube RootThe cube root of a number is the value that, used as a factor three times, gives that number. It is written 3√x. Because 4 × 4 × 4 = 64, 3√64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: 3√−27 = −3.Number System · Grade 8, Algebra 1, Algebra 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 7ReciprocalThe 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 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 5Laws 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 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 7RadicalA radical is an expression that uses the root symbol √, such as √25 or 3√x. The number under the symbol is the radicand, and the small number in the notch is the index, which tells which root to take. With no index, it is a square root: √25 = 5.Number System · Grade 8, Algebra 1, Algebra 2EvaluateEvaluate 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 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 2Negative 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 1Rational NumberA rational number is any number that can be written as a fraction of two integers, ab, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 34, −5 = −51, 1.6 and 0.333… all qualify. Numbers such as √2 and π are irrational.Number System · Grade 7, Grade 8, Algebra 1ExponentAn 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 8FractionA 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 5Exponential 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, PrecalculusLogarithmA logarithm is the exponent you need to raise a base to in order to get a given number. logbx = y means by = x, so log2 32 = 5 because 25 = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Functions · Algebra 2, Precalculus