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Fourth Roots Calculator

Fourth Roots Calculator
\[ \sqrt[4]{x} = \; ? \]
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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:

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

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 \( \sqrt[4]{-81} \) are:

\( 2.12132034 + 2.12132034i \)
\( -2.12132034 + 2.12132034i \)
\( -2.12132034 - 2.12132034i \)
\( 2.12132034 - 2.12132034i \)

Further Reading

Dr Math The Fourth Root of -1

Dr Math DeMoivre's Theorem

De Moivre’s Theorem and Applications



 

Cite this content, page or calculator as:

Furey, Edward "Fourth Roots Calculator"; from http://www.calculatorsoup.com - Online Calculator Resource.

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