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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 x1/4.

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

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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 = a1/4.

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 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.

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

RadicandSymbol4th Root
0⁴√00
1⁴√1±1
2⁴√2±1.189207115
3⁴√3±1.316074013
4⁴√4±1.4142135624
5⁴√5±1.4953487812
6⁴√6±1.5650845801
7⁴√7±1.6265765617
8⁴√8±1.6817928305
9⁴√9±1.7320508076
10⁴√10±1.77827941
11⁴√11±1.8211602868
12⁴√12±1.8612097182
13⁴√13±1.8988289221
14⁴√14±1.9343364203
15⁴√15±1.9679896713
16⁴√16±2
17⁴√17±2.0305431849
18⁴√18±2.0597671439
19⁴√19±2.0877976299
20⁴√20±2.1147425269
21⁴√21±2.1406951429
22⁴√22±2.1657367707
23⁴√23±2.1899387031
24⁴√24±2.2133638394
25⁴√25±2.2360679775
26⁴√26±2.2581008644
27⁴√27±2.279507057
28⁴√28±2.3003266338
29⁴√29±2.3205957871
30⁴√30±2.3403473193
31⁴√31±2.3596110618
32⁴√32±2.37841423
33⁴√33±2.3967817269
34⁴√34±2.4147364028
35⁴√35±2.4322992791
36⁴√36±2.4494897428
37⁴√37±2.4663257146
38⁴√38±2.4828237962
39⁴√39±2.4989993994
40⁴√40±2.5148668594
41⁴√41±2.5304395344
42⁴√42±2.545729895
43⁴√43±2.560749602
44⁴√44±2.5755095769
45⁴√45±2.5900200641
46⁴√46±2.6042906871
47⁴√47±2.6183304987
48⁴√48±2.6321480259
49⁴√49±2.6457513111
50⁴√50±2.6591479485
51⁴√51±2.6723451178
52⁴√52±2.6853496143
53⁴√53±2.6981678764
54⁴√54±2.7108060108
55⁴√55±2.7232698153
56⁴√56±2.7355647997
57⁴√57±2.7476962051
58⁴√58±2.7596690211
59⁴√59±2.7714880025
60⁴√60±2.7831576837
61⁴√61±2.7946823927
62⁴√62±2.8060662633
63⁴√63±2.8173132473
64⁴√64±2.8284271247
65⁴√65±2.8394115144
66⁴√66±2.8502698828
67⁴√67±2.8610055526
68⁴√68±2.871621711
69⁴√69±2.8821214171
70⁴√70±2.8925076085
71⁴√71±2.9027831082
72⁴√72±2.9129506302
73⁴√73±2.9230127857
74⁴√74±2.9329720877
75⁴√75±2.9428309564
76⁴√76±2.9525917237
77⁴√77±2.9622566377
78⁴√78±2.9718278662
79⁴√79±2.9813075013
80⁴√80±2.9906975624
81⁴√81±3
82⁴√82±3.0092166984
83⁴√83±3.0183494793
84⁴√84±3.027400104
85⁴√85±3.0363702767
86⁴√86±3.0452616465
87⁴√87±3.05407581
88⁴√88±3.0628143136
89⁴√89±3.0714786556
90⁴√90±3.0800702882
91⁴√91±3.0885906194
92⁴√92±3.0970410147
93⁴√93±3.1054227991
94⁴√94±3.1137372585
95⁴√95±3.1219856414
96⁴√96±3.1301691601
97⁴√97±3.1382889927
98⁴√98±3.1463462836
99⁴√99±3.1543421455
100⁴√100±3.1622776602

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.

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