 Online Calculators

# Fourth Roots Calculator

Fourth Roots Calculator.
$\sqrt{x} = \; ?$

## Calculator Use

Use this calculator to find the fourth root of a number. It accepts inputs of real numbers for the radicand. This online calculator is set up specifically to calculate 4th root. To calculate any root of a number use our Nth Root Calculator.

For complex or imaginary solutions use Simplify Radical Expressions Calculator.

## Fourth Roots

• Fourth root of 1 is ±1
• Fourth root of 16 is ±2
• Fourth root of 81 is ±3
• Fourth root of 256 is ±4
• Fourth root of 625 is ±5
• Fourth root of 1296 is ±6
• Fourth root of 2401 is ±7
• Fourth root of 4096 is ±8
• Fourth root of 6561 is ±9
• Fourth root of 10000 is ±10

## De Moivre's Theorem

for k = 0, 1, ..., n-1

$$\sqrt[n]{1} = cos\dfrac{2k\pi}{n} + sin\dfrac{2k\pi}{n} \, i$$
$$\sqrt[n]{-1} = cos\dfrac{(2k+1)\pi}{n} + sin\dfrac{(2k+1)\pi}{n} \, i$$

## Fourth Root of a Negative Number

Find the fourth root of negative 81 with n=4 for the 4th root.

Solution:

$$\sqrt{-81}$$
$$= \; \sqrt{81} \cdot \sqrt{ -1 }$$
$$= \; 81^{\frac{1}{4}} \cdot (-1)^{\frac{1}{4}}$$

Using DeMoivre's Theorem we get the equation

$$\small{= 81^{\frac{1}{4}} \cdot \left(cos\left(\dfrac{(2k+1)\pi}{4}\right) + sin\left(\dfrac{(2k+1)\pi}{4}\right)i\right)}$$

Solving our equation for k=0 to k=n-1 (for k = 0, 1, 2 and 3);

The roots of $$\sqrt{-81}$$ are:

$$2.12132034 + 2.12132034i$$
$$-2.12132034 + 2.12132034i$$
$$-2.12132034 - 2.12132034i$$
$$2.12132034 - 2.12132034i$$