Math
Nth Root Calculator
Choose a root index n, enter a radicand, and optionally a power m to evaluate x^(m/n).
The calculator returns an exact answer when the radicand is a perfect power, a simplified radical otherwise, the decimal value, and the equivalent rational-exponent form. Odd roots of negative numbers are real and handled; even roots of negatives are flagged as non-real.
2 for square root, 3 for cube root, and so on.
The number under the root. Whole numbers get simplified radical form.
Set m to evaluate x^(m/n), the m-th power of the root. Keep 1 for a plain root.
Try an example
Result
Result
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- Decimal
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- Exact form
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- Rational exponent form
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- Perfect power?
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Student quick launch
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Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the nth root calculator solves
Choose a root index n, enter a radicand, and optionally a power m to evaluate x^(m/n). The calculator returns an exact answer when the radicand is a perfect power, a simplified A root expression written with the √ symbol, such as a square root. otherwise, the decimal value, and the equivalent rational-The small raised number saying how many times to multiply the base by itself. form. Odd roots of negative numbers are real and handled; even roots of negatives are flagged as non-real.
Formula
How to use it
- Enter the root index n (2 = square root, 3 = cube root, 4 = fourth root, and so on).
- Enter the radicand x. Whole numbers are simplified exactly; decimals give a decimal result.
- Change the power m to evaluate x^(m/n), for example m = 3 and n = 4 for 81^(3/4).
- Read the The answer kept as a fraction or radical instead of a rounded decimal., the decimal, and the rational-exponent notation, then follow the steps.
How to read the answer
The n-th root of x is the number that, raised to the power n, returns x. If the exact form has no radical sign, x was a perfect n-th power. Otherwise the simplified radical is exact and the decimal is an approximation. Raising the root to the power m and taking the root of x^m give the same answer, which is why the rational exponent m/n can be read either way.
Common mistakes and edge cases
- Taking an even root of a negative number as if it were real: ∜-16 is not -2, because (-2)⁴ = +16.
- Simplifying with the wrong group size: for a fourth root you need four identical primes to pull one out.
- Mixing up x^(1/n) with x/n: 81^(1/4) is 3, not 20.25.
- Reading the decimal as exact. ∜2 ≈ 1.189207 is A number that cannot be written as a fraction of two whole numbers, like π or √2..
- Forgetting that a negative power means a The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5.: 8^(-2/3) = 1/4.
Worked examples
Fourth root of 81
81 = 3⁴.
Result
3
Fractional power 81^(3/4)
(∜81)³ = 3³ = 27.
Result
27
Simplify ∜48
48 = 2⁴ × 3, so ∜48 = 2∜3.
Result
2∜3 ≈ 2.632148
Fifth root of a negative
(-2)⁵ = -32.
Result
-2
Negative exponent 8^(-2/3)
1/(∛8)² = 1/4.
Result
1/4
Even root of a negative
Not real.
Result
Error
Decimal radicand
∜2.5 ≈ 1.2574.
Result
1.257433
Frequently asked questions
What is an nth root?+
The n-th root of x is the number y such that y^n = x. The square root is the case n = 2, the cube root n = 3, and so on. In exponent notation it is x^(1/n).
How do I calculate 81 to the 3/4 power?+
Take the fourth root first, then cube: ∜81 = 3 and 3³ = 27. So 81^(3/4) = 27. Doing the power first gives the same answer: ∜(81³) = ∜531441 = 27.
Can I take the fifth root of a negative number?+
Yes. Odd roots of negatives are real because a negative number raised to an odd power stays negative: the fifth root of -32 is -2.
Why is the fourth root of -16 not -2?+
(-2)⁴ = +16, not -16. No real number raised to an even power is negative, so even roots of negatives are imaginary.
How do I simplify a fourth root?+
Factor the radicand into primes and pull out one factor for every four identical primes. ∜48 = ∜(2⁴ × 3) = 2∜3.
What does a negative fractional exponent mean?+
x^(-m/n) = 1 / x^(m/n). For example 8^(-2/3) = 1/(∛8)² = 1/4.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026