Fourth Roots Calculator

How the Fourth Roots Calculator Works

1. Input the Radicand: Enter the number $x$ (called the radicand) into the input field. This is the number for which you want to find the fourth root.

   • Positive values: The calculator will return the positive fourth root of the number.
   • Negative values: If a negative radicand is entered, the calculator will handle it properly by returning the negative fourth root.

2. Negation: You can toggle between positive and negative values using the “+/−” button.

3. Result: After clicking “Calculate,” the calculator will provide:

   • An exact result if $x$ is a perfect fourth power (i.e., the fourth root is an integer).
   • An approximate result rounded to six decimal places if the fourth root is not an integer.

4. Clear: Click the “Clear” button to reset the input fields and result.

Formula for Fourth Roots

The general formula to compute the fourth root of a number $x$ is:

$$\sqrt[4]{x} = x^{1/4}$$

This means that finding the fourth root of $x$ is equivalent to raising $x$ to the power of $\frac{1}{4}$ (or multiplying it by itself four times to get back to $x$).

Examples

1. Perfect Fourth Root Example
   Find the fourth root of $16$.

   Using the formula:

   $$\sqrt[4]{16} = 16^{1/4} = 2$$

   This is because $2 \times 2 \times 2 \times 2 = 16$, so the fourth root of 16 is exactly 2.

2. Approximate Fourth Root Example
   Find the fourth root of $50$.

   Using the formula:

   $$\sqrt[4]{50} \approx 50^{1/4} \approx 2.659$$

   The calculator will return approximately 2.659148 as the fourth root of 50, rounded to six decimal places.

3. Negative Fourth Root Example
   Find the fourth root of $-81$.

   Using the formula:

   $$\sqrt[4]{-81} = -\sqrt[4]{81} = -3$$

   This is because $-3 \times -3 \times -3 \times -3 = -81$. Hence, the fourth root of $-81$ is $-3$.