Calculate the fourth root of any number. View perfect fourth powers, compare with square root, explore all complex roots, and verify results instantly.
The fourth root of a number x, denoted ⁴√x, is the value r such that r⁴ = x. A handy shortcut: the fourth root is simply the square root of the square root, since ⁴√x = √(√x). This property makes fourth roots particularly intuitive and useful across many areas of mathematics and applied science.
Fourth roots arise in geometry (the relationship between a hypercube's hyper-volume and edge length), statistics (fourth moments and kurtosis), and signal processing (RMS of RMS calculations). Unlike cube or fifth roots, the real fourth root is only defined for non-negative numbers — but every non-zero number has exactly four complex fourth roots equally spaced around the complex plane.
This calculator computes the real fourth root with adjustable precision, shows the equivalence to √(√x), lists all four complex roots, compares with other root indices, and provides a reference table of perfect fourth powers. Check the example with realistic values before reporting.
Fourth roots are more common in practical applications than many people realize. The Stefan-Boltzmann law in physics states that radiated power is proportional to T⁴, so finding the temperature from radiation requires a fourth root. In statistics, the fourth central moment (kurtosis) involves fourth powers, and its root helps normalize distributions. In computer graphics, color space conversions often use power-of-4 relationships.
This calculator provides instant fourth-root computation, complex root visualization, comparison with other root indices, and a reference table — everything you need in a single tool.
Fourth Root: ⁴√x = x^(1/4) = √(√x) Complex roots: x^(1/4) · e^(iπk/2), k = 0, 1, 2, 3 Verification: (⁴√x)⁴ = x
Result: 5
The fourth root of 625 is 5 because 5⁴ = 5 × 5 × 5 × 5 = 625. Equivalently, √(√625) = √25 = 5.
The fourth root operation is the inverse of raising to the fourth power. Because ⁴√x = x^(1/4), it sits naturally within the family of fractional exponents. A useful identity is ⁴√x = √(√x), which means you can compute fourth roots using only a square-root button. For manual estimation, Newton's method gives rapid convergence: rₙ₊₁ = (3rₙ + x/rₙ³)/4.
Every non-zero complex number z has exactly four fourth roots, given by |z|^(1/4) · e^(i(θ+2πk)/4) for k = 0, 1, 2, 3, where θ = arg(z). For positive reals, the four roots are ±r and ±ri, where r is the positive real fourth root. For negative reals, the four roots are rotated by 45°. Understanding these roots is essential in control theory, electrical engineering, and quantum mechanics.
In thermodynamics, the Stefan-Boltzmann law P = εσAT⁴ means solving for temperature requires the fourth root of the power ratio. In structural engineering, fourth-power relationships appear in beam deflection formulas. In finance, some option-pricing models use fourth-moment corrections that involve fourth roots for normalization.
The fourth root of x is the number r where r⁴ = x. It equals x raised to the power 1/4, or equivalently the square root of the square root of x.
Not in real numbers. The fourth root of a negative number requires complex numbers. This calculator shows all four complex roots when you select that mode.
⁴√x = √(√x). You can compute the fourth root by taking the square root twice. Also, (√x)² = x and (⁴√x)⁴ = x.
Numbers that equal an integer raised to the 4th power: 1, 16, 81, 256, 625, 1296, 2401, 4096, etc. Use this as a practical reminder before finalizing the result.
Every non-zero number has exactly 4 fourth roots in the complex plane, equally spaced at 90° intervals. Positive real numbers have one positive real fourth root.
Geometry (4D volumes), statistics (kurtosis calculations), signal processing, computer graphics (gamma-related transforms), and physics (Stefan-Boltzmann law relates temperature to radiation via T⁴). Keep this note short and outcome-focused for reuse.