Fifth Root Calculator

What is a Fifth Root?

A fifth root of a number is a value that, when multiplied by itself five times, gives the original number. In other words, it's the reverse operation of raising a number to the fifth power. For any real number \(x\), the fifth root of \(x\) is the number \(y\) such that \(y^5 = x\).

How to Calculate a Fifth Root

Calculating a fifth root can be done through various methods:

• Using a calculator with a fifth root function
• Applying the exponent rule: \(\sqrt[5]{x} = x^{\frac{1}{5}}\)
• Using prime factorization for perfect fifth powers
• Employing numerical methods like Newton's method for approximation

Formula

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

\[ y = \sqrt[5]{x} \]

Which is equivalent to:

\[ y^5 = x \]

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

Calculation Steps

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

Example

Let's calculate the fifth root of 32:

1. We want to find \(y\) such that \(y^5 = 32\)
2. We recognize that \(2 \times 2 \times 2 \times 2 \times 2 = 32\)
3. Therefore, \(\sqrt[5]{32} = 2\)
4. To verify: \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\), which confirms our result

Thus, the fifth root of 32 is 2.

Visual Representation