Last updated: Mar 14, 2024

Fifth Root Calculator

Table of contents

How to use this fifth root calculator Exponents and Roots Calculating 5th root of a real number Other calculators FAQs

Our fifth root calculator is a quick and easy tool to help you find the fifth root of a real number. It also lets you calculate other roots, too!

In the following article, we shall quickly learn what roots are and how to calculate the fifth root of a number, complete with examples.

How to use this fifth root calculator

Using this fifth root calculator is rather simple:

Enter the number a for which you wish to calculate the fifth root.
The fifth root calculator will immediately show you the fifth root of this number.

For example, say you want to find the 5ᵗʰ root of 7776. Enter 7776 in the a field. Right away, we get the result as 6.

We can use this calculator to find other roots by providing a different value for the n parameter. For example, to find the third root of the same number 7776, change n from 5 to 3. The calculator will instantly tell us the 3ʳᵈ root of 7776 is 19.81.

❗ Our calculator will not allow you to find the n^th-root for a negative number if n is even, as that would give a complex root.

The following sections will explain the math behind this calculator, so we recommend you keep reading.

Exponents and Roots

The exponent operation raises a number $b$ to a given exponent (or power) $n$ , which is the same as multiplying the number with itself $n$ times:

a = b^n = b \times b \times b \times ...n \text{ times}

For example, 2 raised to the power 5 is:

2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32

The root operation is essentially the inverse of the exponent operation — It finds the number $b$ that must be raised to the power $n$ to get $a$ .

a = b^n \Lrarr b = \sqrt[n]{a}

Therefore, we can say that 2 is the fifth root of 32:

2^5 = 32 \implies \sqrt[5]{32} = 2

Calculating 5th root of a real number

Let's look at how to calculate the fifth root of the number 7776:

Prime factorize the number. To learn how to do this quickly, head to our prime factorization calculator.

\qquad \begin{align*} 7776 =& 2 \times 2 \times 2 \times 2 \times 2\\ &\times 3 \times 3 \times 3 \times 3 \times 3 \end{align*}

Group these prime factors by sets of 5.

\qquad 7776 = 2^5 \times 3^5

Now, take the fifth root of the number using these prime factors:

\qquad \sqrt[5]{7776} = \sqrt[5]{2^5 \times 3^5} = 2\times 3 = 6

Not every problem is as simple as that, of course! For example, using the same steps to find the fifth root of 125 will leave us at a stalemate at step 3, with $\sqrt[5]{125} = \sqrt[5]{5^3}$ . That's where this fifth root calculator comes in! By simply entering 125 in a, we find that $\sqrt[5]{125} = 2.6265$ .

Other calculators

If you enjoyed our fifth root calculator, you may find these calculators useful, too:

Root calculator; and
Fourth root calculator.

FAQs

How do you calculate the fifth root of a number?

The following steps will help you calculate the fifth root of a number:

Prime factorize the number.
Group these prime factors by sets of 5.
Take the fifth root of the number using these prime factors.

If you have difficulty calculating this manually, use our fifth root calculator instead!

What is the fifth root of 7962624?

The fifth root of 7962624 is 24. To find this answer, follow these steps:

Prime factorize the number 7962624:
7962624 = 2¹⁵ × 3⁵.
Group these prime factors by sets of 5:
7962624 = (2³)⁵ × 3⁵ .
Take the fifth root of 7962624 using these prime factors:
⁵√(7962624) = ⁵√((2³)⁵ × 3⁵) = 2³ × 3 = 24.

You can verify this result using our fifth root calculator.

$\qquad \qquad \sqrt[n]{a}$

a

n

5^th root

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