Compute the real fourth root of any non-negative number instantly — with step-by-step working that shows the double square-root method.
The fourth root of a number is the value that, multiplied by itself four times, returns the original number. Written ∜x, it answers the question “what number to the fourth power equals x?” The fourth root of 625 is 5, because 5 × 5 × 5 × 5 = 625. The fourth root of 10,000 is 10. This calculator finds the real fourth root of any non-negative number and shows the classic two-step working: take a square root, then take the square root again.
Fourth roots sit one level above square roots in the family of radicals, and they appear wherever fourth powers arise. In physics, the Stefan–Boltzmann law relates a star’s temperature to the fourth root of its radiated power; in engineering, beam-deflection formulas contain fourth roots of stiffness ratios; in statistics, some normalizations raise data to fractional powers near one-fourth. Students meet fourth roots when simplifying radical expressions like ∜(256x8) and when solving biquadratic equations of the form ax4 + c = 0. Because most calculators bury the fourth root two menus deep, having a dedicated tool with visible steps saves real time.
One important boundary first: there is no real fourth root of a negative number. Raising any real number to the fourth power gives a non-negative result, so the reverse question has no real answer for negative inputs. (In complex numbers there are four answers, but this calculator stays in the real numbers, where school and engineering problems live.) Enter zero or any positive number and the calculator returns the principal — non-negative — fourth root.
Type any number greater than or equal to zero into the input box and press “Calculate Fourth Root.” The widget first validates your input: blanks, non-numbers, and negatives each get a clear explanation instead of a cryptic error. For a valid input it displays the fourth root in large type, then walks through the double square-root method — ∜x = √(√x) — showing both intermediate square roots, and finishes with a verification step that raises the answer to the fourth power so you can see it reconstruct your input.
Try 625 first: the steps show √625 = 25, then √25 = 5, then the check 54 = 625. Then try a non-perfect value like 100: the calculator returns about 3.1623, with steps √100 = 10 and √10 ≈ 3.1623, and the check 3.16234 ≈ 100. The “Reset” button restores 625. Results are rounded to at most 8 decimal places, with very large or very small values shown in scientific notation.
A fourth root is a special case of an n-th root: ∜x is the non-negative number r such that r4 = x. The small 4 tucked into the radical’s crook is called the index, and it tells you how many self-multiplications the answer must survive. Because the index 4 is even, the principal fourth root is defined only for non-negative radicands and is itself non-negative — the fourth root of 16 is 2, not ±2, even though (−2)4 also equals 16. The equation x4 = 16 has two real solutions, ±2, but the function ∜16 returns the single principal value 2.
The double square-root identity is the key structural fact: ∜x = √(√x). It holds because (x1/2)1/2 = x1/4 by the power-of-a-power rule. This is not just a curiosity — it is genuinely how fourth roots were computed before electronic calculators, and it is why the widget’s steps look the way they do. It also explains the relationship between fourth roots and squares: ∜(x2) = √x for x ≥ 0, since (x2)1/4 = x1/2.
Fourth roots grow very slowly. While √10,000 = 100, ∜10,000 is only 10; and ∜1,000,000 is just 10·∜10 ≈ 31.6. Each factor-of-10,000 increase in x adds only 10 to the fourth root. This slow growth is exactly why fourth roots tame explosive quantities in formulas: they compress huge ranges into manageable numbers. The flip side is precision sensitivity — because the function is so flat for large x, small errors in the root correspond to large errors in x4, so the widget’s verification step matters.
Notation varies by context. Pure mathematics writes ∜x; applied work often writes x1/4 or x^(1/4); programmers write pow(x, 0.25) or x**0.25. All mean the principal real fourth root. Be careful with negative bases in software: (-16)**0.25 in Python returns a complex number, while many calculators simply throw an error — both are “correct” within their number systems, which is why this widget states its real-numbers boundary up front.
∜x = x1/4 = √(√x), for x ≥ 0
Check: (∜x)4 = x
The formula has two faces. The exponent form x1/4 is what you type into a calculator or spreadsheet (in Excel, =POWER(A1,1/4) — with parentheses around 1/4, since =A1^1/4 would compute (A1^1)/4). The nested-radical form √(√x) is the hand-computation method and the widget’s step-by-step display. Both rest on the exponent law (xa)b = xab: applying the one-half power twice gives the one-fourth power.
The check line is the definition running backwards and doubles as your error detector: whatever the widget returns, raising it to the fourth power must reproduce the input (up to rounding). If it does not, something went wrong. Related identities worth keeping nearby: ∜(x4) = |x| for real x, and ∜(xy) = ∜x · ∜y for non-negative x and y, which lets you factor radicands — ∜(162) = ∜(81 × 2) = 3∜2 ≈ 3.567.
Example 1 — ∜625. First square root: √625 = 25. Second square root: √25 = 5. Check: 54 = 625. So ∜625 = 5. Recognizing 625 = 54 is a standard fluency target; the perfect fourth powers 1, 16, 81, 256, 625, 1,296, 2,401 (14 through 74) are worth memorizing the way multiplication tables are.
Example 2 — ∜100 (not a perfect fourth power). √100 = 10, then √10 ≈ 3.16228. Check: 3.162284 ≈ 100.0000. Notice that ∜100 = √10 — a special case of ∜(x2) = √x with x = 10. Whenever the radicand is a perfect square, the fourth root collapses to an ordinary square root.
Example 3 — Simplifying ∜(2,401x4). Factor: 2,401 = 74. So ∜(74x4) = ∜((7x)4) = |7x| = 7|x|. For x ≥ 0 this is simply 7x. This is the standard algebra-textbook move: pull perfect fourth powers out of the radical.
Example 4 — Solving x4 = 5,000. Take the fourth root of both sides: x = ±∜5,000 ≈ ±8.409. The ± appears because we solved an equation (two real solutions), whereas the function ∜5,000 returns only the positive 8.409. Keeping the function/equation distinction straight prevents the most common sign error with even roots.
Astronomy — stellar temperatures. The Stefan–Boltzmann law says a star’s total radiated power scales with the fourth power of its surface temperature: P ∝ T4. If one star radiates 16 times the power per unit area of another, it is only ∜16 = 2 times as hot. Fourth roots convert observed brightness ratios into temperature ratios — the reason a modest temperature difference produces a dramatic luminosity difference.
Structural engineering — beam deflection. The deflection of a beam under load involves the fourth power of its length and the inverse fourth power appears when solving for required stiffness. When an engineer needs the beam depth that limits sag to a target, fourth roots turn a deflection requirement into a dimension. Getting the root right to three significant figures is standard practice because the fourth power amplifies small errors.
Audio and signal processing. Decibel scales are logarithmic, and converting between power ratios and amplitude ratios involves square roots; cascaded conversions and certain filter designs introduce fourth-root relationships. Audio engineers meeting “twice as loud needs ten times the power” are one square root away from fourth-root territory in multi-stage gain calculations.
Finance — annualized growth. An investment that grows by a factor of 1.5 over 4 years has an annualized growth rate of ∜1.5 − 1 ≈ 10.7% per year. Any “per-period rate from a multi-period total” problem is an n-th root problem, and the 4-year case is a fourth root. The widget computes ∜1.5 ≈ 1.10668 directly.
Writing ∜(−x) as a real number. There is no real fourth root of a negative number. If your problem produces ∜(−81), either the problem expects a complex answer (the four complex fourth roots of −81) or, far more likely, a sign error happened earlier. The widget refuses negatives precisely to surface this.
Confusing ∜x with √x. The index matters enormously: √625 = 25 but ∜625 = 5. Students who misread the tiny 4 as decoration get answers that are squares of the correct value. Always check which root the problem asks for before computing.
Forgetting the ± when solving equations. The function ∜16 = 2 (principal root only), but the equation x4 = 16 has solutions x = ±2. Dropping the negative solution loses half the answer set. Use the calculator for the principal root, then add ± yourself when solving equations.
Distributing roots over sums. ∜(a + b) is not ∜a + ∜b. For instance, ∜(16 + 65) = ∜81 = 3, but ∜16 + ∜65 ≈ 2 + 2.84 = 4.84. Roots distribute over multiplication and division, never over addition or subtraction.
Spreadsheet order of operations. In Excel, =625^1/4 computes (625^1)/4 = 156.25, not the fourth root. You must write =625^(1/4) or =POWER(625,1/4). This single missing pair of parentheses is one of the most common computational errors in applied work.
Continue with roots and powers: the index-2 square root this tool builds on (∜x = √(√x)); the Cube Root Calculator covers odd-index roots where negatives are allowed; the Exponent Calculator computes arbitrary powers including fractional ones; the Cube Calculator inverts cube roots; and the Scientific Calculator offers general-purpose power functions.
The real fourth root ∜x is the principal (non-negative) value r with r4 = x, defined for x ≥ 0; equivalently x1/4 or √(√x). No real fourth root exists for x < 0. Key identities: (∜x)4 = x; ∜(x4) = |x|; ∜(xy) = ∜x∜y for x,y ≥ 0. The equation x4 = a (a > 0) has real solutions x = ±∜a. Applications: Stefan–Boltzmann temperature ratios, beam deflection, annualized 4-year growth rates.
A fourth root of x is a number that equals x when multiplied by itself four times. The principal fourth root, written ∜x, is the non-negative such number: ∜625 = 5 because 5 × 5 × 5 × 5 = 625. It is the inverse operation of raising to the fourth power.
5. Since 625 = 54 (5 × 5 × 5 × 5 = 625), the principal fourth root ∜625 = 5. Via the double square-root method: √625 = 25, then √25 = 5.
Take two square roots in a row: ∜x = √(√x). For ∜2,0736… for example ∜1,296: √1,296 = 36, then √36 = 6, so ∜1,296 = 6. For non-perfect values like ∜100, take √100 = 10 and then √10 ≈ 3.1623 using any square-root method.
The principal fourth root ∜x is never negative — it is defined as the non-negative root. However, the equation x4 = 16 has two real solutions, 2 and −2, because (−2)4 = 16. So ∜16 = 2 (function value), while x4 = 16 gives x = ±2 (equation solutions).
There is no real fourth root of a negative number, because no real number raised to the fourth power is negative. In the complex numbers, a negative number has four distinct fourth roots. Unless your course has introduced complex numbers, treat ∜(negative) as undefined and look for a sign error upstream.
Both are themselves: ∜1 = 1 and ∜0 = 0. Like all roots, the fourth root fixes 0 and 1, since 04 = 0 and 14 = 1. These are useful sanity checks for any fourth-root computation.
Use =POWER(A1,1/4) or =A1^(1/4) — the parentheses around 1/4 are essential, because =A1^1/4 is interpreted as (A1^1)/4. In Google Sheets the same formulas work identically.
The perfect fourth powers begin 1, 16, 81, 256, 625, 1,296, 2,401, 4,096, 6,561, 10,000 — that is, 14 through 104. Memorizing the first seven (up to 2,401) makes simplifying fourth-root expressions dramatically faster.
Cite this page as: SigmaCalculator. (2026). Fourth Root Calculator. https://sigmacalculator.net/math-calculators/fourth-root-calculator/
Last updated: October 9, 2026