4th Root

In math, the 4th root b of a number a is such that b⁴ = a. By definition, when you multiply b by itself 4 times you get the value of a.

A 4-th root is usually denoted ⁴√x, but it can also be written in exponential form with the base x and the exponent 1/4: x^1/4 or x^1/4.

Read on to learn everything about these numbers, including the properties, and make sure to check out our calculator.

Calculator

Number:

4th Root:

±

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If you happen to know exponentiation, then you can think of the 4th root of a number as the inverse operation to elevating a number to the power of 4.

Definition

Whereas in exponentiation elevating a number a to the power of 4 is defined as a⁴ = b, the 4th root b is defined as b = a^1/4.

For example with a = 2401 we get:
=

=

=

=

In other words, the 4th root of 2401 is 7, because 7 times 7 times 7 times 7 is 7.

Keep reading to learn an important property of the fourth root.
On this Site You Can Find all About 4th Roots, Including a Calculator! Click To Tweet If the index n of a root is is even, such as in the case here with n = 4, then every positive number
has two such roots:

The positive is also known as principal 4th root

and the negative 4th root
Together, they are written as
Let’s have a look the previous example:

=

=

=

=
Proof: -7⁴ = 2401 73⁴ = 2401

Both values multiplied 4 x by itself = 2401.

Make sure to understand that multiplying the 4th root four times by itself produces the original number (not the 4th root x 4).

The positive root is always the principal.

This should sound familiar if you have read our square root article – the home page.

Next, we explain how the parts are called. Keep reading to learn all about the topic.

Parts

As depicted, the parts of the fourth root are:



The radix sign, which tells us that it is a mathematical root, and the index of 4, which tells us that it is the 4-th root.

The number below the radix, x, is the radicand.

The result of the mathematical operation is denoted by the equation sign and called the root.

4th Root Symbol

The symbol √ is called radical sign, or radix.

Ahead is a few words regarding our calculator.

About our 4th Root Calculator

Our calculator at the top of this page computes the fourth root of any non-negative real number.

Just enter a valid radicand; you then automatically obtain both, the principal root as well as the negative result.

Note that our tool works both ways; that is the math is bidirectional.

Use the up and down arrows (called spinners) to increase or decrease the input value.

You can change both, the upper as well as the lower input field of our 4-th root calculator.

If this online app has been of use to you bookmark it now. Next, we discuss the properties.

4-th Root Properties

With and , the properties are as follows:

• = -a if a < 0 and a if ≥ 0

The most important property is the first; the negative number tends to be forgotten. Read on to see the examples:

4th Root Examples

We use the list of properties above to show you some examples in the order of appearance:

• 
  

Frequently searched terms on this site include:

• 4th Root of of 81
• 4th Root of 16
• 4th Root of 8

4th Root in Excel

In Excel you enter the syntax for

in a cell, like this:

=POWER(radicand,1/4)

For example, to calculate

insert =POWER(64,1/4).

In the next section we explain how to do the math.

How to Calculate the 4th Root

A very efficient procedure for extracting the fourth root is the Newton–Raphson method, also known as Newton’s method detailed below:

You begin with a guessed starting value and then iterate the steps until you’re happy with the precision.



In the most basic version of the method, f is a single-variable function and f′ its derivative.

You may think of it as the 4th root formula.

Next are FAQs in the context of this article.

Frequently Asked Questions

Click on the question which is of interest to you to see the collapsible content answer.

What is the 4th Root Called?

The 4th root is called fourth root.

How Do You Type a 4th Root on a Computer?

On a Windows computer hit the Alt key and type 8732 simultaneously, on a Mac hit Option and type 221C.

How Do You Do 4th Roots on a Calculator?

Look for the power function, then insert the 4th root as x^(1/4).

If something about fourth roots needs clarification do not hesitate getting in touch with us.

We are constantly aiming to improve this site, and truly appreciate your feedback!

Websites with calculators which are related to the root of a number can be found in the “recommended sites” section” in the sidebar.

Table of Fourth Roots

Ahead is the wrap up of our information regarding the 4th root of a number.

4th Root Summary

You have made it to the end of our article about the 4 th root.

The 4th root b of a number a is such that b⁴ = a; b = ±⁴√a.

⁴√ is the 4-th root symbol consisting of the radix and the index 4.

Note that you can find the fourth root of many numbers by using the search form in the header menu.

If you are happy with our information, then please share our content on your favorite social media site.

And don’t forget to install our free PWA app (see menu or sidebar) called square root.

For comments and anything else you might have about 4th sqrt fill in the designated form below.

You might also be interested that we have a nth root calculator which accepts all defined indexes (n) for members of the real number set R.

We aim to reply as soon as possible.

Thanks for visiting our article 4th Root.

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