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 4th root is usually denoted \sqrt[4]{x}, but it can also be written in exponential form with the base a and the exponent 1/4: x^{\frac{1}{4}}.

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

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:

\sqrt[4]{2401} = \sqrt[4]{7^{4}} = 7^\frac{4}{4} = {7^1} = 7

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.
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If the index n of a root is is even, such as in the case here with n = 4, then every positive number \in \mathbb{R} has two such roots:

The positive is also known as principal 4th root \sqrt[n]{x}, and the negative 4th root -\sqrt[n]{x}.

Together, they are written as \pm\sqrt[4]{x}.

Let’s have a look the previous example:

\sqrt[4]{2401} = \sqrt[4]{7^{4}} = 7^\frac{4}{4} = {7^1} = \pm 7

Proof:

-7⁴ = 2401

73⁴ = 2401

The positive root is always the principal.

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 equal sign and called the root.

4th Root Symbol

The symbol √ is called radical sign, or radix.

Ahead is our calculator.

4th Root Calculator

Our calculator 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.

If this app has been of use to you bookmark it now.

Next, we discuss the properties.

4th Root Properties

With a,b \in \mathbb{R^{+}} and k,m \in \mathbb{N}, the properties are as follows:

• \sqrt[4]{x} = |a| = -a if a < 0 and a if ≥ 0
• \sqrt[4]{ab}= \sqrt[4]{a} \sqrt[4]{b}
• \sqrt[4]{a/b}= \frac{\sqrt[4]{a}}{\sqrt[4]{b}}
• \sqrt[4]{0} = 0
• \sqrt[4]{a^{-m}} = \frac{1}{\sqrt[4]a^{m}}
• \sqrt[4k]{a^{m}} = \sqrt[4]{a^{km}}
• \sqrt[4]{\sqrt[m]{a}} = \sqrt[4m]{a} = \sqrt[m]{\sqrt[4]{a}}

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:

• \sqrt[4]{4096} = \pm 8
• \sqrt[4]{20736} = \sqrt[4]{81} \sqrt[4]{256} = 3 x 4 = \pm 12
• \sqrt[4]{256/16}= \frac{\sqrt[4]{256}}{\sqrt[4]{16}} = 16/2 = \pm 8
• \sqrt[4]{5^{-4}} = \frac{1}{\sqrt[4]{5^{4}}} = \frac{1}{5} = \pm 1/5
• \sqrt[4]{9^{4}} = \sqrt{9^{2}} = \sqrt{81} = \pm 9
• \sqrt[4]{\sqrt[3]{125}} = \sqrt[4 x 3]{125} = \pm \sqrt[12]{125}
• \sqrt[4]{\sqrt[m]{6}} = \sqrt[4m]{5} = \sqrt[m]{\sqrt[4]{6}}

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 x^{\frac{1}{4}} in a cell, like this:

=POWER(radicand,1/4)

For example, to calculate \sqrt[4]{64} 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.

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

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 frequently asked questions in the context of this article.

FAQs

Table of Fourth Roots

Ahead is the bottom line.

Conclusion

You have made it to the end of our article.

Note that you can find the value of many 4th roots by using the search form in the header menu.

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More Information:

• n-th Root