Fourth Root Calculator

Calculate the fourth root of any number with step-by-step verification and alternate representations.

Enter a Number

Quick values:

Result

Fourth Root of 81
3
&sup4;√81 = 3
Input (x) 81
Fourth Root 3
As Square Root of Square Root √(√81) = √9 = 3
Verification (result&sup4;) 3&sup4; = 81
Is Perfect Fourth Power? Yes

Step-by-Step Solution

&sup4;√x = x^(1/4) = √(√x)

What Is a Fourth Root?

The fourth root of a number x is a value that, when raised to the fourth power, equals x. Mathematically, the fourth root of x is written as 4√x or x1/4. For example, the fourth root of 81 is 3, because 34 = 3 × 3 × 3 × 3 = 81.

Methods for Calculating Fourth Roots

Exponent Method

Raise the number to the power of 1/4.

4√x = x1/4

Double Square Root

Take the square root twice in succession.

4√x = √(√x)

Perfect Fourth Powers

Numbers whose fourth roots are integers.

1, 16, 81, 256, 625, 1296...

Negative Numbers

Fourth roots of negative numbers are not real (require complex numbers).

4√(-x) = not real

Perfect Fourth Powers

A perfect fourth power is a number that can be expressed as an integer raised to the fourth power. The first several perfect fourth powers are:

  • 14 = 1
  • 24 = 16
  • 34 = 81
  • 44 = 256
  • 54 = 625
  • 64 = 1,296
  • 74 = 2,401
  • 84 = 4,096
  • 94 = 6,561
  • 104 = 10,000

Relationship to Other Roots

The fourth root is closely related to the square root. Taking the fourth root is equivalent to taking the square root twice: 4√x = √(√x). This relationship can simplify calculations, especially when working with perfect squares.

Applications

  • Physics: calculating certain decay rates and wave frequencies.
  • Engineering: stress analysis and material science formulas.
  • Statistics: fourth moment calculations (kurtosis).
  • Computer graphics: gamma correction and color space transformations.