Fourth Roots Calculator

What is a Fourth Root?

A fourth root of a number is a value that, when multiplied by itself four times, gives the original number. In other words, it's the reverse operation of raising a number to the fourth power. For any real number \(x\), the fourth root of \(x\) is the number \(y\) such that \(y^4 = x\).

How to Calculate a Fourth Root

Calculating a fourth root can be done through various methods:

• Using a calculator with a fourth root function
• Applying the exponent rule: \(\sqrt[4]{x} = x^{\frac{1}{4}}\)
• Using prime factorization for perfect fourth powers
• Employing numerical methods like Newton's method for approximation

Formula

The formula for the fourth root of a number \(x\) is:

\[ y = \sqrt[4]{x} \]

Which is equivalent to:

\[ y^4 = x \]

Where \(x\) is the number we're finding the fourth root of, and \(y\) is the result.

Calculation Steps

1. Identify the number \(x\) for which you want to calculate the fourth root
2. If \(x\) is a perfect fourth power, find the number that, when raised to the fourth power, equals \(x\)
3. If \(x\) is not a perfect fourth power, use a calculator or computational method to find \(\sqrt[4]{x}\)
4. For complex numbers, there are four fourth roots. Find all four if necessary
5. Verify your result by raising it to the fourth power, which should equal the original number \(x\)

Example

Let's calculate the fourth root of 16:

1. We want to find \(y\) such that \(y^4 = 16\)
2. We recognize that \(2 \times 2 \times 2 \times 2 = 16\)
3. Therefore, \(\sqrt[4]{16} = 2\)
4. To verify: \(2^4 = 2 \times 2 \times 2 \times 2 = 16\), which confirms our result

Thus, the fourth root of 16 is 2.

Visual Representation