Compute the 5th root of any number instantly — enter a value and get ⁵√x exactly, with perfect-power detection and step-by-step working. Free, fast, and mobile-friendly.
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The fifth root of a number x, written ⁵√x, is the number that gives x when raised to the 5th power. So ⁵√32 = 2, because 2⁵ = 32. Fifth roots undo fifth powers exactly the way square roots undo squares — except for one friendly difference: because 5 is odd, negative numbers have real fifth roots too (⁵√(−32) = −2). Our free Fifth Roots Calculator evaluates ⁵√x for any input instantly, detects perfect fifth powers, and shows every step.
Fifth roots appear wherever something compounds five times. The average annual growth rate over five years is the fifth root of the total growth factor, minus one — the 5-year CAGR that investors quote constantly. In geometry, the side length of a 5-dimensional hypercube with a given hypervolume is a fifth root. Signal processing and statistics meet fifth roots in power-law scaling and moment calculations.
Because most fifth roots are irrational (⁵√100 ≈ 2.511886432), hand computation means tedious approximation. This calculator evaluates xˡ⁵ directly, checks whether your number is a perfect fifth power (like 32, 243, or 100000), verifies the answer by raising it back to the 5th power, and handles negative inputs correctly — no complex numbers, no errors, just the real root.
By definition, y = ⁵√x means y⁵ = x. Equivalently, ⁵√x = xˡ⁵, which is how the calculator evaluates it. The fifth root is the inverse function of the fifth power: applying one undoes the other, so (⁵√x)⁵ = x and ⁵√(y⁵) = y for every real x and y. This clean inversion holds for all reals precisely because the function t → t⁵ is strictly increasing — it never turns around.
That monotonicity is why negative inputs work. Squaring folds negatives onto positives (so √(−4) has no real answer), but fifth powers preserve sign: (−2)⁵ = −32. Running the film backward, ⁵√(−32) = −2 with no ambiguity. Every odd root — cube roots, fifth roots, seventh roots — shares this property, while every even root is restricted to non-negative inputs.
Perfect fifth powers are the integers whose fifth root is also an integer: 0, ±1, ±32, ±243, ±1,024, ±3,125, ±7,776, ±16,807, ±32,768, ±100,000, … (1⁵, 2⁵, 3⁵, …). They thin out fast — by 10⁵ the gaps are enormous — so nearly every integer you meet has an irrational fifth root. The calculator tests the rounded result by raising it to the 5th power, catching perfect powers exactly.
Fifth roots also connect to logarithms: ⁵√x = e^(ln(x)/5) for x > 0. Before electronic calculators, this identity was the practical route — look up ln(x), divide by 5, exponentiate. It also reveals the growth story: fifth roots grow very slowly, since raising to the 1/5 power compresses huge ranges (⁵√100,000 = 10, but ⁵√3,200,000 ≈ 20).
⁵√x = xˡ⁵, with ⁵√(−x) = −⁵√x
Logarithm form (x > 0): ⁵√x = e^(ln(x)/5) — the classical hand-computation route.
Recognize 32 as a power of 2: 32 = 2⁵. Therefore ⁵√32 = 2 exactly. Verify: 2⁵ = 32. This is the cleanest fifth root after ⁵√1 = 1, and the calculator flags it as a perfect fifth power.
243 = 3⁵ (3² = 9, 3³ = 27, 3⁴ = 81, 3⁵ = 243). So ⁵√243 = 3 exactly. Powers of 3 grow fast — the next perfect fifth power, 4⁵ = 1024, is already past a thousand.
Since 10⁵ = 100,000, the root is exactly 10. In general, ⁵√(10^(5k)) = 10^k: fifth roots strip five zeros at a time, which makes estimating easy — ⁵√3,200,000 is near ⁵√10^6·³² ≈ 20.
Because 5 is odd, the negative comes straight through: ⁵√(−32) = −⁵√32 = −2. Check: (−2)⁵ = −32. No complex numbers needed — this is the real fifth root.
An investment grows from $10,000 to $16,105 over 5 years. What was the average annual return? The total growth factor is 1.6105, and the annualized rate is ⁵√1.6105 − 1 ≈ 1.10 − 1 = 10% per year. The fifth root converts a 5-year total into the steady yearly rate — this is the 5-year CAGR every fund fact sheet reports.
A data-center architect is told a 5-dimensional parameter grid must hold exactly 7,776 configurations with equal divisions along each axis. The divisions per axis are ⁵√7776 = 6 (since 6⁵ = 7776) — a perfect fifth power, so the grid is exactly 6×6×6×6×6. The calculator’s perfect-power detection confirmed the clean answer instantly.
A biologist models a population that multiplies by the same factor each generation for 5 generations, growing 40-fold overall. The per-generation multiplier is ⁵√40 ≈ 2.0915 — the population slightly more than doubles each generation. The fifth root splits total growth into equal per-period steps, the standard move in any compounding analysis.
How would you compute ⁵√100 by hand to six decimals? Newton’s method iterates x_{n+1} = x_n − (x_n⁵ − a)/(5x_n⁴), which simplifies to x_{n+1} = (4x_n + a/x_n⁴)/5. Start at x_0 = 3 (since 3⁵ = 243 > 100): x_1 = (12 + 100/81)/5 ≈ 2.6469, x_2 ≈ 2.5217, x_3 ≈ 2.51208, x_4 ≈ 2.5118865 — correct digits roughly double each round. Four iterations, six decimals.
The iteration works because each step follows the tangent line of t⁵ − a down to the axis, and near the root the tangent hugs the curve. Convergence is quadratic: once close, precision doubles per step. The only danger is a bad start (x_0 = 0 divides by zero) or an even root of a negative — neither troubles fifth roots of reals.
Modern libraries actually compute xˡ⁵ the way this calculator does: via exp(ln(x)/5) with careful argument reduction, refined by one Newton polish. The direct pow() call hides all of this — but knowing Newton’s method means you can reproduce the digit 2.511886432 on paper with nothing but long division and patience.
100 sits between 2⁵ = 32 and 3⁵ = 243, closer to 32. Newton’s method from 2.5 converges to 2.511886432. Verify: 2.511886432⁵ ≈ 100.0000000. The calculator shows this bracketing so the size is checkable at a glance.
Write 0.00032 = 32 × 10⁻⁵. Then ⁵√0.00032 = ⁵√32 × 10⁻¹ = 2 × 0.1 = 0.2. Check: 0.2⁵ = 0.00032. Fifth roots scale cleanly with powers of 10⁵, just like square roots do with 100.
An index rises from 400 to 900 in 5 years. Growth factor = 900/400 = 2.25. Annualized rate = ⁵√2.25 − 1 ≈ 1.17608 − 1 = 17.6% per year. Sanity check: 1.17608⁵ ≈ 2.25. Investors compare funds on exactly this number.
A curious pattern: the units digit of n⁵ always equals the units digit of n (2⁵ = 32 ends in 2, 7⁵ = 16807 ends in 7). This is a mini Fermat’s little theorem — n⁵ ≡ n (mod 10) — and it means the last digit of a perfect fifth power reveals its root’s last digit for free. ⁵√16,807 must end in 7; combined with 16,807 sitting between 6⁵ = 7776 and 7⁵ = 16807, the root is exactly 7. Two observations, zero computation.
Compute ⁵√1024 exactly (hint: factor 1024 as a power of 2), then verify with the calculator.
A $5,000 investment becomes $8,000 in 5 years. Find the annualized return using the fifth-root CAGR formula, and check that compounding your answer 5 times returns 1.6.
Explain why ⁵√(−100,000) is a real number but √(−100,000) is not. What is the exact value of the fifth root?
A fund advertises “12% average annual returns over the last 5 years.” An analyst checks: 1.12⁵ = 1.7623, so $10,000 should have become about $17,623. The statement shows growth from $10,000 to only $15,500 — a factor of 1.55.
The true annualized return is ⁵√1.55 − 1. The calculator gives ⁵√1.55 ≈ 1.0916, so the real CAGR is about 9.2% — not 12%. The fund had averaged the five yearly percentages arithmetically, a classic error: arithmetic means overstate compounded growth.
One fifth root exposed a 2.8-percentage-point exaggeration. The analyst’s report recommended the geometric (fifth-root) figure as the honest headline — and the client moved their money. Roots are not abstract here; they are the difference between marketing and math.
The Fifth Roots Calculator evaluates ⁵√x = xˡ⁵ for any real input, keeping the correct negative sign for negative inputs (odd roots are real). It shows the fractional-exponent step, verifies by raising the answer to the 5th power, detects perfect fifth powers exactly (verifying candidate⁵ equals the input), and brackets non-perfect inputs between consecutive fifth powers.
The fifth root of x, written ⁵√x, is the number that equals x when raised to the 5th power. For example, ⁵√32 = 2 because 2⁵ = 32. Equivalently, ⁵√x = xˡ⁵.
Yes. Because 5 is odd, fifth powers preserve sign, so odd roots of negatives are real: ⁵√(−32) = −2. Only even roots (square, fourth) of negatives are non-real.
Exactly 2, since 2⁵ = 32. Similarly ⁵√243 = 3, ⁵√1024 = 4, and ⁵√100,000 = 10 — these are perfect fifth powers.
Divide the final value by the initial value to get the growth factor G, then compute ⁵√G − 1. If $10,000 grows to $16,105 in 5 years, G = 1.6105 and the CAGR is ⁵√1.6105 − 1 ≈ 10% per year.
Yearly percentages compound multiplicatively, so the honest average is the geometric mean — the 5th root of the total growth factor. Arithmetic averaging overstates growth; in the case study it inflated 9.2% into a claimed 12%.
Bracket it between consecutive fifth powers (2⁵ = 32 < 100 < 243 = 3⁵, so ⁵√100 ≈ 2.5), then refine with Newton’s iteration x ← (4x + a/x⁴)/5, which doubles correct digits each round.
Everything: ⁵√32 = 2, while 32/5 = 6.4. The fifth root asks “what to the 5th power gives x,” while x/5 just divides. They coincide only at x = 0.
Yes — perfect fifth powers thin out extremely fast (after 10⁵ = 100,000 the gaps are enormous), so nearly every integer has an irrational fifth root. An answer like 2.511886432 for ⁵√100 is correct, not a rounding artifact.
Cite this page as: Fifth Roots Calculator. Sigma Calculator. Retrieved October 9, 2026, from https://sigmacalculator.net/math-calculators/fifth-roots-calculator/
Last updated: October 9, 2026