n-th Root

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

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

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

Definition

Whereas in exponentiation elevating a number a to the power of n is defined as an = b, the n-th root b is defined as b = a1/n.

For example with a = 15625 and n = 3 we get:

\sqrt[3]{15625} = \sqrt[3]{5^{6}} = 5^\frac{6}{3} = {5^2} = 25

In other words, the 3rd root of 15625 is 25, because 25 times 25 times 25 is 15625.

Observe that the index n is 3 (odd). Thus, 25 is the only real 3rd root. Keep reading to learn an important property in case n is even.

On this Site You Can Find all About n-th Roots, Including a Calculator! Click To Tweet If n is even, then every positive number \in \mathbb{R} has two such roots, the positive also known as principal n-th root \sqrt[n]{x}, and the negative n-th root -\sqrt[n]{x}.

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

Let’s have a look this example when the index is even:

\sqrt[8]{6561} = \sqrt[8]{3^{8}} = 3^\frac{8}{8} = 3^1 = \pm 3

Proof:

-38 = 6561

+38 = 6561

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 root are:

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

The number below the radix is the radicand.

The result of the mathematical operation is denoted by the equal sign and called the root. Parts of n-th Root

n-th Root Symbol

The symbol √ is called radical sign, or radix.

Ahead is our calculator.

n-th Root Calculator

Our calculator computes the root of any non-negative real number.

Just enter a valid index and radicand; you then automatically obtain both, the principal root as well as the negative result (if applicable).

\sqrt[n]{x} Calculator

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Next, we discuss the properties.

n-th Root Properties

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

  • \sqrt[n]{x} = |a| = -a if a < 0 and a if ≥ 0
  • \sqrt[n]{ab}= \sqrt[n]{a} \sqrt[n]{b}
  • \sqrt{a^{m}} = a^{\frac{m}{2}} \Rightarrow \sqrt{a} = a^{\frac{1}{2}}
  • \sqrt[n]{a/b}= \frac{\sqrt[n]{a}}{\sqrt[n]{b}}
  • \sqrt[2]{a} = \sqrt{a}
  • \sqrt[n]{0} = 0
  • \sqrt[n]{a^{-m}} = \frac{1}{\sqrt[n]a^{m}}
  • \sqrt[nk]{a^{m}} = \sqrt[n]{a^{km}}
  • \sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a} = \sqrt[m]{\sqrt[n]{a}}

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

n-th Root Examples

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

  • \sqrt[6]{729} = \pm 3
  • \sqrt[5]{32768} = \sqrt[5]{32} \sqrt[5]{1024} = 2 x 4 = 8
  • \sqrt{5^{2}} = 5^{\frac{2}{2}} = 5^{1} = 5
  • \sqrt[3]{216/343}= \frac{\sqrt[3]{216}}{\sqrt[3]{343}} = 6/7
  • \sqrt[2]{3^{-2}} = \frac{1}{\sqrt[2]{3^{2}}} = \frac{1}{\sqrt{9}} = 1/3
  • \sqrt[n]{8^{n2}} = \sqrt{8^{2}} = \sqrt{64} = 8
  • \sqrt[7]{\sqrt[3]{125}} = \sqrt[7x3]{125} = \sqrt[21]{125}
  • \sqrt[n]{\sqrt[m]{5}} = \sqrt[nm]{5} = \sqrt[m]{\sqrt[n]{5}}

Frequently searched terms on this site include:

nth Root in Excel

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

=POWER(radicand,1/index)

For example, to calculate \sqrt[3]{64} insert =POWER(64,1/3).

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

How to Calculate the n-th Root

A very efficient procedure for extracting the 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 of it as the n-th root formula.

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

FAQs

How do You Solve for the n-th Root?

If y = x^n, then the n-th root x = y^1/n.

How do You Multiply n-th Roots?

\sqrt[n]{xy}= \sqrt[n]{x} \sqrt[n]{y}

How do You Write the n-th Root on a Computer?

Use a Latex editor, enter \sqrt[]{} and put the index in the square brackets and the radicand in the curly brackets.

What is the n-th Root of a Real Number?

The n-th root of a real number x is any number y whose n-th power is x.



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Conclusion

You have made it to the end of our article.n-th RootNote that you can find the value of many n-th roots by using the search form in the header menu.

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