What is a fourth root?

The fourth root of a number x is the value that, multiplied by itself four times, gives x. Written [4]{x}, it undoes raising to the fourth power: since 3^4 = 81, the fourth root of 81 is 3, i.e. [4]{81} = 3. It can also be thought of as a square root of a square root: [4]{x} = {{x}}.

What is the fourth root of a negative number?

There's no real number that, raised to the fourth power, gives a negative value (a negative number to an even power is always positive), so the fourth root of a negative number is complex, not real. Just like a square root, [4]{-16} has no real value — it is expressed using the imaginary unit i as 2i.

Why does a positive number have two real fourth roots, but the radical symbol gives only one?

Both 3 and -3 raised to the fourth power equal 81, since (-3)^4 = 81 too. But the radical symbol [4]{\,} is defined to return only the non-negative one — the **principal root**. So [4]{81} = 3, not -3, even though (-3)^4 = 81 as well. If a problem needs both real fourth roots, they are written explicitly as [4]{81} = 3.

What is a perfect fourth power?

A perfect fourth power is a whole number that is some integer raised to the fourth power — 1, 16, 81, 256, 625, 1296, … Their fourth roots come out as exact whole numbers ([4]{256} = 4) rather than an irrational decimal.

Why isn't the fourth root of 48 just a decimal?

Because 48 is not a perfect fourth power, [4]{48} is an irrational number — its decimal expansion never ends or repeats (2.632148…). The exact value is the simplified radical 2[4]{3}, since 48 = 16 × 3 and [4]{16} = 2 can be pulled out from under the root. The decimal is only an approximation; the radical form is exact.

How do you simplify a fourth root by hand?

Break the number into prime factors, then pull out every complete group of four. For [4]{48}: 48 = 2^4 × 3, which has one complete group of four 2s and a single leftover 3. Pulling the group of four out gives 2[4]{3}.

What is the inverse of a fourth root?

Raising to the fourth power is the inverse of taking a fourth root: ([4]{x})^4 = x for x 0. If you know [4]{y} = x, then x^4 = y. Use the [Exponents calculator](/en/calculators/algebra/exponents) to compute x^4 directly.

Fourth Root Calculator

Find the fourth root of any number as an exact simplified radical (like 2⁴√3) plus a decimal value, including perfect fourth powers and negative inputs.

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What a fourth root is

The fourth root of a number $x$ is the value that, multiplied by itself four times, produces $x$ :

$\sqrt[4]{x} = y \quad \text{means} \quad y \times y \times y \times y = x$

Since $3 \times 3 \times 3 \times 3 = 81$ , the fourth root of $81$ is $3$ — written $\sqrt[4]{81} = 3$ . A fourth root can also be seen as a square root nested inside another square root: $\sqrt[4]{x} = \sqrt{\sqrt{x}}$ , since taking a square root twice divides the exponent by $2$ twice, or by $4$ overall.

The principal root: why ⁴√81 = 3, not −3

Both $3$ and $-3$ raised to the fourth power equal $81$ , since $(-3)^4 = (-3)\times(-3)\times(-3)\times(-3) = 81$ too. But the radical symbol is defined to always return the non-negative root, called the principal fourth root. So $\sqrt[4]{81} = 3$ by convention — if a problem needs both real fourth roots, they are written explicitly as $\pm\sqrt[4]{81} = \pm 3$ . This calculator, like every fourth root symbol, always shows the principal (non-negative) root for a positive input.

Perfect fourth powers vs. simplified radicals

A perfect fourth power is a whole number that is some integer raised to the fourth power — $1, 16, 81, 256, 625, 1296, \ldots$ Its fourth root is an exact whole number:

$x$	$\sqrt[4]{x}$
$16$	$2$
$81$	$3$
$256$	$4$
$625$	$5$

Most numbers are not perfect fourth powers, so their fourth root is irrational — a never-ending, non-repeating decimal. Rather than truncate that decimal, the exact answer is written as a simplified radical: a whole-number coefficient times the smallest possible root. For example, $\sqrt[4]{48} = 2\sqrt[4]{3}$ , because $48 = 16 \times 3$ and $\sqrt[4]{16} = 2$ can be pulled out from under the root, leaving $3$ — which has no fourth-power factors left to extract.

Tip: A radical is "fully simplified" when the number left under the root has no perfect-fourth-power factors other than 1.

\sqrt[4]{3}

is already simplified (3 is prime);

2\sqrt[4]{3}

can't be simplified any further.

Simplifying a radical, step by step

To simplify $\sqrt[4]{48}$ by hand:

Factor into primes: $48 = 2^4 \times 3$ .
Group into fours: the $2^4$ is a complete group of four; the single $3$ has no group to join.
Pull each group out as its fourth root: $2^4 \to 2$ , which becomes the coefficient.
Leave the leftover factor under the root: the unpaired $3$ stays inside, giving $\sqrt[4]{48} = 2\sqrt[4]{3}$ .

This calculator shows that same prime-factorization breakdown for whatever number you enter.

Negative numbers and the imaginary unit i

No real number raised to the fourth power gives a negative value, since an even power always produces a non-negative result. So the fourth root of a negative number isn't a real number — like a square root, it's complex. The imaginary unit $i$ is defined as $i = \sqrt{-1}$ , which lets any negative fourth root be written exactly:

$\sqrt[4]{-16} = \sqrt[4]{16} \times \sqrt[4]{-1} = 2i$

The same simplification rules apply to the positive part before the $i$ is attached — for example $\sqrt[4]{-48} = 2\sqrt[4]{3}\,i$ , since $\sqrt[4]{48}$ simplifies to $2\sqrt[4]{3}$ first. Unlike a real result, a complex fourth root has no single real decimal approximation, so this calculator shows only the exact form for negative inputs.

The inverse: raising to the fourth power

Taking a fourth root and raising to the fourth power undo each other: $(\sqrt[4]{x})^4 = x$ for $x \geq 0$ . If $\sqrt[4]{y} = x$ , then $x^4 = y$ — so $\sqrt[4]{625} = 5$ and $5^4 = 625$ are the same fact, read in opposite directions. Use the Exponents calculator to compute $x^4$ directly.

Worked examples

Example 1 — perfect fourth power: $\sqrt[4]{16} = 2$ , exactly, since $2 \times 2 \times 2 \times 2 = 16$ .

Example 2 — simplified radical: $\sqrt[4]{48} = 2\sqrt[4]{3} \approx 2.632148$ , since $48 = 16 \times 3$ .

Example 3 — negative input: $\sqrt[4]{-16} = 2i$ , a complex result, since no real number raised to the fourth power gives $-16$ .

Example 4 — already simplified: $\sqrt[4]{3} \approx 1.316074$ stays as $\sqrt[4]{3}$ — 3 has no fourth-power factors to pull out.

Where fourth roots show up

Fourth roots appear in statistics and physics formulas involving fourth-power relationships (such as certain radiation and diffusion laws), and in geometry when reversing a fourth-power scaling relationship back to a linear one. They are also the natural next step after square roots and cube roots in the general pattern of $n$ -th roots — see the Square Root calculator for the even-index case at $n=2$ , where negative inputs likewise become complex instead of real, and the general Radicals calculator for any index.

Frequently asked questions

What is a fourth root?: The fourth root of a number $x$ is the value that, multiplied by itself four times, gives $x$ . Written $\sqrt[4]{x}$ , it undoes raising to the fourth power: since $3^4 = 81$ , the fourth root of $81$ is $3$ , i.e. $\sqrt[4]{81} = 3$ . It can also be thought of as a square root of a square root: $\sqrt[4]{x} = \sqrt{\sqrt{x}}$ .
What is the fourth root of a negative number?: There's no real number that, raised to the fourth power, gives a negative value (a negative number to an even power is always positive), so the fourth root of a negative number is complex, not real. Just like a square root, $\sqrt[4]{-16}$ has no real value — it is expressed using the imaginary unit $i$ as $2i$ .
Why does a positive number have two real fourth roots, but the radical symbol gives only one?: Both $3$ and $-3$ raised to the fourth power equal $81$ , since $(-3)^4 = 81$ too. But the radical symbol $\sqrt[4]{\,}$ is defined to return only the non-negative one — the **principal root**. So $\sqrt[4]{81} = 3$ , not $-3$ , even though $(-3)^4 = 81$ as well. If a problem needs both real fourth roots, they are written explicitly as $\pm\sqrt[4]{81} = \pm 3$ .
What is a perfect fourth power?: A perfect fourth power is a whole number that is some integer raised to the fourth power — $1, 16, 81, 256, 625, 1296, \ldots$ Their fourth roots come out as exact whole numbers ( $\sqrt[4]{256} = 4$ ) rather than an irrational decimal.
Why isn't the fourth root of 48 just a decimal?: Because $48$ is not a perfect fourth power, $\sqrt[4]{48}$ is an irrational number — its decimal expansion never ends or repeats ( $2.632148\ldots$ ). The exact value is the simplified radical $2\sqrt[4]{3}$ , since $48 = 16 \times 3$ and $\sqrt[4]{16} = 2$ can be pulled out from under the root. The decimal is only an approximation; the radical form is exact.
How do you simplify a fourth root by hand?: Break the number into prime factors, then pull out every complete group of four. For $\sqrt[4]{48}$ : $48 = 2^4 \times 3$ , which has one complete group of four 2s and a single leftover 3. Pulling the group of four out gives $2\sqrt[4]{3}$ .
What is the inverse of a fourth root?: Raising to the fourth power is the inverse of taking a fourth root: $(\sqrt[4]{x})^4 = x$ for $x \geq 0$ . If you know $\sqrt[4]{y} = x$ , then $x^4 = y$ . Use the [Exponents calculator](/en/calculators/algebra/exponents) to compute $x^4$ directly.