How to Take The Fourth Root Without A Calculator

Calculating the fourth root of a number is a common mathematical operation that can be performed without a calculator using several different methods. This guide explains the concept of the fourth root, provides step-by-step methods for manual calculation, and includes a built-in calculator for quick reference.

What is the Fourth Root?

The fourth root of a number is a value that, when raised to the power of 4, gives the original number. In mathematical terms, if y is the fourth root of x, then:

Formula

y = x^(1/4)

For example, the fourth root of 16 is 2 because 2 × 2 × 2 × 2 = 16. The fourth root is related to the square root, which is the second root. The fourth root is essentially the square root of the square root.

Methods for Calculating Fourth Root

There are several methods to calculate the fourth root without a calculator. Here are three common approaches:

Method 1: Using Square Roots

First, find the square root of the number.
Then, find the square root of the result from step 1.
The final result is the fourth root of the original number.

Example

To find the fourth root of 16:

Square root of 16 = 4
Square root of 4 = 2
Fourth root of 16 = 2

Method 2: Using Exponents and Logarithms

Take the natural logarithm of the number.
Divide the result by 4.
Exponentiate the result to get the fourth root.

Formula

y = e^(ln(x)/4)

Method 3: Using the Babylonian Method (Approximation)

Make an initial guess for the fourth root.
Improve the guess using the formula: new_guess = (3 × old_guess + x/(old_guess³)) / 4
Repeat the process until the result is accurate enough.

Note

This method is iterative and may require several steps for accuracy, especially for non-perfect fourth powers.

Worked Example

Let's calculate the fourth root of 81 using the square root method:

First square root: √81 = 9
Second square root: √9 ≈ 3.000
Final result: 3.000

Therefore, the fourth root of 81 is approximately 3.000. For verification, 3 × 3 × 3 × 3 = 81.

FAQ

What is the difference between the fourth root and the square root?

The fourth root of a number is the value that, when raised to the power of 4, gives the original number. The square root is the value that, when raised to the power of 2, gives the original number. The fourth root is essentially the square root of the square root.

Can the fourth root of a negative number be calculated?

No, the fourth root of a negative number cannot be calculated using real numbers. The fourth root of a negative number is not a real number but an imaginary number.

Is there a difference between the fourth root and the cube root?

Yes, the fourth root is different from the cube root. The fourth root is the value that, when raised to the power of 4, gives the original number, while the cube root is the value that, when raised to the power of 3, gives the original number.

How accurate are the manual methods for calculating fourth roots?

The accuracy of manual methods depends on the method used and the number of iterations performed. The square root method is exact for perfect fourth powers, while the Babylonian method provides an approximation that can be refined with more iterations.