Fourth Roots Calculator finds the fourth root of a number, returning an exact integer for perfect fourth powers and a 12-digit decimal approximation otherwise. Because four is an even index, negative inputs have no real fourth root, and the tool reports that case as undefined rather than guessing.
Find the fourth root of a perfect fourth power
A perfect fourth power equals some integer raised to the fourth power. 2^4 = 16, so the fourth root of 16 is exactly 2. Fourth Roots Calculator searches for an integer whose fourth power matches the input and returns that integer directly whenever a match exists, without rounding.
Work through the calculation for 81
To confirm the fourth root of 81 is 3, raise 3 to the fourth power: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81. Four multiplications, or equivalently squaring twice (3^2 = 9, then 9^2 = 81), land exactly on 81.
Fourth Roots Calculator displays this as "81 is 3^4" alongside the returned root.
Approximate a fourth root that is not exact
Most integers are not perfect fourth powers. The fourth root of 50 sits between the fourth root of 2^4 = 16 and the fourth root of 3^4 = 81, since 50 falls between those two values; the exact result is approximately 2.659.
Fourth Roots Calculator computes this using the fractional exponent x^(1/4) and rounds to 12 significant digits for display.
Take a fourth root as two square roots
Because 4 = 2 × 2, the fourth root of a number is the same as taking the square root twice in a row: the fourth root of x equals the square root of (the square root of x).
For x = 81, the square root of 81 is 9, and the square root of 9 is 3, matching the direct fourth-root result. This double-square-root shortcut works for any nonnegative number and is often faster by hand than estimating a fourth root directly.
Handle negative inputs correctly
Fourth roots are an even-index root, so a negative input has no real fourth root, since any real number raised to the fourth power produces a nonnegative result. The fourth root of −16 is undefined over the real numbers, even though −2 raised to the fourth power looks tempting to test: (−2)^4 = 16, a positive result, not −16.
Fourth Roots Calculator returns a clear error for negative inputs rather than a wrong or misleading value.
Avoid this common mistake
Assuming every root behaves like an odd-index root and accepts negative inputs freely leads to errors here. The fourth root is even-indexed, grouped with square roots and sixth roots, and none of those accept negative real inputs.
Confirm whether a root's index is even or odd before assuming a negative input is valid; even indices always require a nonnegative input for a real result.
Simplify a fourth root with a perfect-power factor
When a number is not a perfect fourth power outright but contains one as a factor, the fourth root can still be partially simplified.
The fourth root of 32 factors as the fourth root of (16 × 2), and since 16 = 2^4, this simplifies to 2 times the fourth root of 2, a cleaner exact form than a raw decimal.
Fourth Roots Calculator applies this factoring step whenever a perfect-fourth-power factor exists inside the radicand.
Compare the fourth root to the square root of the square root
Because 4 = 2 × 2, computing a fourth root by taking two nested square roots is often faster by hand than estimating directly.
For x = 625, the square root of 625 is 25, and the square root of 25 is 5, so the fourth root of 625 is 5, matching a direct check: 5^4 = 5 × 5 × 5 × 5 = 625.
This nested approach works because each square root operation divides the exponent by 2, and doing it twice divides by 4 overall.
Frequently asked questions
What is the fourth root of 81?
The fourth root of 81 is 3, since 3 raised to the fourth power (3 × 3 × 3 × 3) equals 81 exactly.
How do you find a fourth root without a calculator?
To find a fourth root without a calculator, take the square root of the number, then take the square root of that result again, since the fourth root equals the square root applied twice.
Can a fourth root be negative?
A fourth root cannot be a negative real number as its result for a nonnegative input, and a negative input itself has no real fourth root at all, since any real number raised to an even power is nonnegative.
What is the fourth root of a negative number?
The fourth root of a negative number is undefined over the real numbers, because raising any real number to the fourth power, an even power, always produces a result of zero or greater.
How is the fourth root related to the square root?
The fourth root is related to the square root by nesting: the fourth root of x equals the square root of the square root of x, since taking a square root twice divides the exponent by 2 twice, for a total division by 4.
What is the fourth root of 16?
The fourth root of 16 is 2, since 2 raised to the fourth power (2 × 2 × 2 × 2) equals 16 exactly.
What is the fourth root of 625?
The fourth root of 625 is 5, since 5 raised to the fourth power (5 × 5 × 5 × 5) equals 625 exactly, which also checks out by taking the square root of 625 (25) and then the square root of 25 (5).
Can a non-perfect fourth power still be partially simplified?
A non-perfect fourth power can still be partially simplified whenever it contains a perfect-fourth-power factor; the fourth root of 32 becomes 2 times the fourth root of 2, since 32 factors as 16 times 2 and 16 is a perfect fourth power.
Summary
Fourth Roots Calculator computes the fourth root of a number, returning an exact integer for perfect fourth powers like 81 (root 3) and a 12-digit decimal approximation otherwise. Because four is an even index, negative inputs return an undefined error rather than a value.
Enter a nonnegative real number to see its exact or approximate fourth root, or use the double-square-root shortcut to check the result by hand.