Fifth Root Calculator

Enter any real number to find its fifth root (⁵√x = x^(1/5)), verify the result, and compare it to the nearest perfect fifth power.

Quick Facts

Formula
⁵√x = x^(1/5)
The fifth root undoes raising a number to the 5th power.
Negative numbers
⁵√(−x) = −⁵√x
Because 5 is odd, every real number has exactly one real fifth root — unlike the square root.
Perfect fifth powers
1⁵=1, 2⁵=32, 3⁵=243, 4⁵=1024, 5⁵=3125
Useful benchmarks for estimating fifth roots by hand.
Logarithm method
x^(1/5) = 10^(log₁₀x / 5)
Works for x > 0 when a root key isn't available.

Your Results

Calculated
Fifth Root
-
⁵√x = x^(1/5)
Verification (root⁵)
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Should match your input x
Nearest Perfect Fifth Power
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Closest whole-number fifth power
Difference from Nearest Power
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x − nearest whole-number⁵

Ready

Enter a number, then press Calculate.

Formula and Method for the Fifth Root

The fifth root of a number x, written ⁵√x or x^(1/5), is the value that equals x when multiplied by itself five times: (⁵√x)⁵ = x. Because 5 is an odd number, the fifth root is defined for every real number, including negative numbers and zero — unlike the square root (an even-index root), which has no real value for negative inputs.

How the calculation works

This calculator raises your number to the power of 1/5. For non-negative x, that is simply x^(1/5). For negative x, it computes −(|x|)^(1/5), since an odd-index root of a negative number is negative — for example, ⁵√(−32) = −2 because (−2)⁵ = −32. The tool also raises the result back to the 5th power so you can verify the answer, and reports the nearest whole-number fifth power for comparison.

Common mistakes

  • Assuming no real root exists for negative numbers: that rule applies to even-index roots (square root, fourth root), not odd-index roots like the fifth root or cube root.
  • Confusing x^(1/5) with x/5 or x⁵: the fifth root divides the exponent, it does not divide the value, and it is the inverse of raising to the 5th power, not the same operation.
  • Rounding too early: for irrational roots (most inputs that are not perfect fifth powers), keep several decimal places until the final step to avoid compounding rounding error.

Checking your result

The fastest check is to raise your answer back to the 5th power and see if you land back on the original number, allowing for rounding. This calculator does that automatically in the "Verification" result. You can also compare your input to the nearest perfect fifth power (1, 32, 243, 1024, 3125, …) to sanity-check the order of magnitude of the root.

Applications

Fifth roots appear in growth-rate problems, such as finding the average annual multiplier over a five-year period from a total growth factor. They also show up in engineering formulas built around fifth-power relationships, and in algebra and precalculus courses when solving equations of the form x⁵ = k or simplifying radical expressions with index 5.

Frequently Asked Questions

What is the fifth root of a number?
The fifth root of a number x, written ⁵√x or x^(1/5), is the value that equals x when multiplied by itself five times. For example, ⁵√243 = 3, because 3 × 3 × 3 × 3 × 3 = 243.
Can you find the fifth root of a negative number?
Yes. Because 5 is an odd number, every real number has exactly one real fifth root, including negative numbers. For example, ⁵√(−32) = −2, because (−2)⁵ = −32. This is different from square roots, where negative numbers have no real square root.
How do you calculate a fifth root without a calculator?
For perfect fifth powers, compare against 1⁵=1, 2⁵=32, 3⁵=243, 4⁵=1024, and 5⁵=3125 to find a whole-number match. For other numbers, use logarithms: x^(1/5) = 10^(log₁₀x / 5), or use the x^(1/y) or x^0.2 function on a scientific calculator.
What is the difference between a fifth root and a square root?
A square root (index 2) is only defined for non-negative real numbers and has two roots (positive and negative) for positive inputs. A fifth root (index 5) is defined for every real number, including negatives, and there is exactly one real fifth root for any real input.

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