6th Root

In math, the 6th root b of a number a is such that b⁶ = a. By definition, when you multiply b by itself 6 times you get the value of a.

A 6th root is usually denoted \sqrt[6]{x}, but it can also be written in exponential form with the base a and the exponent 1/6: x^{\frac{1}{6}}.

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 6th root of a number as the inverse operation to elevating a number to the power of 6.

Definition

Whereas in exponentiation elevating a number a to the power of 6 is defined as a⁶ = b, the 6th root b is defined as b = a^1/6.

For example with a = 262144 we get:

\sqrt[6]{262144 } = \sqrt[6]{8^{6}} = 8^\frac{6}{6} = {8^1} = 8

In other words, the 6th root of 262144 is 8, because 8 times 8 times 8 times 8 times 8 times 8 is 262144.

Keep reading to learn an important property of the sixth root.
On this Site You Can Find all About 6th 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 = 6, then every positive number \in \mathbb{R} has two such roots:

The positive is also known as principal 6th root \sqrt[n]{x}, and the negative 6th root -\sqrt[n]{x}.

Together, they are written as \pm\sqrt[6]{x}.

Let’s have a look at this example:

\sqrt[6]{4096} = \sqrt[6]{4^{6}} = 4^\frac{6}{6} = {4^1} = \pm 4

Proof:

-4⁶ = 4096

4⁶ = 4096

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 sixth root are:

The radix sign, which tells us that it is a mathematical root, and the index of 6, which tells us that it is the 6-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.

6th Root Symbol

The symbol √ is called radical sign, or radix.

Ahead is our calculator.

6th Root Calculator

Our calculator computes the sixth 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.

6th Root Properties

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

• \sqrt[6]{x} = |a| = -a if a < 0 and a if ≥ 0
• \sqrt[6]{ab}= \sqrt[6]{a} \sqrt[6]{b}
• \sqrt[6]{a/b}= \frac{\sqrt[6]{a}}{\sqrt[6]{b}}
• \sqrt[6]{0} = 0
• \sqrt[6]{a^{-m}} = \frac{1}{\sqrt[6]a^{m}}
• \sqrt[6k]{a^{m}} = \sqrt[6]{a^{km}}
• \sqrt[6]{\sqrt[m]{a}} = \sqrt[6m]{a} = \sqrt[m]{\sqrt[6]{a}}

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

6th Root Examples

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

• \sqrt[6]{117649} = \pm 7
• \sqrt[6]{20736} = \sqrt[6]{64} \sqrt[6]{15625} = 2 x 5 = \pm 10
• \sqrt[6]{729/4096}= \frac{\sqrt[6]{729}}{\sqrt[6]{4096}} = 3/4 = \pm 0.75
• \sqrt[6]{8^{-6}} = \frac{1}{\sqrt[6]{8^{6}}} = \frac{1}{8} = \pm 1/8
• \sqrt[12]{5^{4}} = \sqrt[6]{5^{2}} = \sqrt[6]{25} = \pm 1.709975947
• \sqrt[6]{\sqrt[3]{a}} = \sqrt[6 x 3]{a} = \pm \sqrt[18]{a}
• \sqrt[6]{\sqrt[m]{7}} = \sqrt[6m]{7} = \sqrt[m]{\sqrt[6]{7}}

Frequently searched terms on this site include:

• 6th Root of 64
• 6th Root of 729
• 6th Root of 4096

6th Root in Excel

In Excel you enter the syntax for x^{\frac{1}{6}} in a cell, like this:

=POWER(radicand,1/6)

For example, to calculate \sqrt[100]{64} insert =POWER(100,1/6).

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

How to Calculate the 6th Root

A very efficient procedure for extracting the sixth 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 6th root formula.

Next are frequently asked questions in the context of this article.

FAQs

Table of Sixth 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 6th roots 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 app.

For comments and anything else you might have fill in the designated form below. We aim to reply as soon as possible.

Thanks for visiting our website.

More Information:

• n-th Root