How this instrument works
The fourth root of x, written ⁴√x or x^(1/4), is the unique non-negative real number y that satisfies y⁴ = x. Because any real number raised to an even power comes out zero or positive, no real y can ever produce a negative y⁴ — so unlike a fifth root or a cube root, which are odd roots defined for every real number including negatives, a fourth root only exists on the reals when x is zero or positive. This sheet enforces that boundary at the x field rather than silently returning a complex value.
There's a shortcut hidden in the exponent: raising a number to the one-quarter power is the same as taking a square root twice, since (x^(1/2))^(1/2) = x^(1/4). Eighty-one is a clean case to see it with — √81 = 9, and √9 = 3, landing on the same answer this sheet's formula reaches in a single step. That nesting is not a coincidence; it falls straight out of the exponent rule (x^a)^b = x^(ab), with a = b = 1/2 multiplying together to the 1/4 a fourth root demands.
A positive x actually has two real numbers whose fourth power lands on it — 2⁴ and (−2)⁴ both equal 16 — but the radical symbol ⁴√ names only the non-negative one by convention, the same rule an ordinary square root already follows. That restriction resurfaces in physics: the Stefan–Boltzmann law ties a black body's radiated power to the fourth power of its temperature, so recovering a temperature ratio from a measured power ratio means taking a fourth root, and only the positive root is physically meaningful.
- Enter your value into the x field — only zero or positive numbers are accepted, since a fourth root has no real value below zero.
- Read the result in the ⁴√x field: the single non-negative number that, raised to the fourth power, rebuilds x exactly.
- Try x = 16 to see ⁴√x = 2, then check it by hand: 2 × 2 × 2 × 2 comes to 16.
- Adjust x and watch ⁴√x update instantly — useful for reversing any fourth-power relationship, from a power ratio to a scaled length.
Worked example — a star twice as hot
A reference star's surface radiates a fixed power per unit area, and a second star's surface radiates exactly 16 times as much. The Stefan–Boltzmann law ties radiated power to temperature through a fourth power, P is proportional to T⁴, so the temperature ratio is the fourth root of the power ratio: set x = 16 and this sheet returns ⁴√x = 2. The second star runs twice as hot as the reference, not sixteen times — the fourth power compresses a large power ratio into a modest temperature ratio.
The arithmetic checks cleanly in reverse: 2⁴ = 2 × 2 × 2 × 2 = 16, matching the power ratio exactly, with no rounding anywhere in the chain. The nested-root shortcut confirms it a second way — √16 = 4, and √4 = 2 — landing on the identical answer the direct formula returns in a single step.
Questions
What is a fourth root?
The fourth root of x is the unique non-negative real number y that satisfies y⁴ = x. ⁴√16 = 2 because 2 × 2 × 2 × 2 = 16, and it is written x^(1/4) or ⁴√x interchangeably — the small 4 marking which root, distinct from the unmarked √ that always means a square root.
Why does a negative number have a real cube or fifth root, but no real fourth root?
Squaring a negative number flips it positive, and squaring again keeps it positive, so any real y raised to the fourth power lands at zero or above — no real y can ever reach a negative target. Cube roots and fifth roots are odd roots: an odd number of negative factors keeps the result negative, so −8 and −32 have real cube and fifth roots. A fourth root, being even, has no such escape route.
Does 16 really have two fourth roots?
Yes, among the real numbers: 2⁴ and (−2)⁴ both equal 16. But the radical symbol ⁴√ is defined to return only the principal, non-negative root by convention — the same rule an ordinary square root follows, where √16 means 4, never −4. This sheet always returns that principal value.
Is there a shortcut for finding a fourth root by hand?
Take a square root twice. Because (x^(1/2))^(1/2) = x^(1/4), the fourth root of 81 comes out the same way whether computed directly or as two nested square roots: √81 = 9, then √9 = 3. The trick only lands on a whole number when the intermediate square root does too, but it's a fast mental check on any calculator's answer.
Where does a fourth-power relationship show up outside a math class?
The Stefan–Boltzmann law in physics is the clearest example: a black body's radiated power per unit area scales with the fourth power of its absolute temperature. Reversing that relationship — recovering a temperature ratio from a measured power ratio — is exactly a fourth-root calculation, the same one this sheet performs.
What happens if I enter a negative number into this calculator?
The x field only accepts zero or positive values, and a negative entry is flagged rather than silently producing a wrong or complex result. That restriction reflects real mathematics, not a limitation of this sheet: no real number raised to the fourth power can ever equal a negative value.