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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 6-th root is usually denoted 6√x, but it can also be written in exponential form with the base x and the exponent 1/6: x^1/6 or x1/6.

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 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 = a1/6.

For example with a = 262144 we get:

=


=


=


=


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.
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If the index n of a root is is even, such as in the case here with n = 6, then every positive number has two such roots:

The positive is also known as principal 6th root

and the negative 6th root

Together, they are written as

Let’s have a look at this example:

=


=


=


=


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 above 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 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:

6th 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:

6th Root in Excel

In Excel you enter the syntax for

in a cell, like this:

=POWER(radicand,1/6)

For example, to calculate

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.

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



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

What is the 6th Root Called?

The 6th root is called sixth root.

How Do You Type a 6th Root on a Computer?

Write the 6 as ⁶ followed by the radix sign √: ALT + 8310 + ALT 251.

How Do You do 6th Roots on a Calculator?

Look for the power function, then insert the 6th root as x^(1/6).



Table of Sixth Roots

RadicandSymbol6th Root
0⁶√00
1⁶√1±1
2⁶√2±1.1224620483
3⁶√3±1.2009369552
4⁶√4±1.2599210499
5⁶√5±1.307660486
6⁶√6±1.3480061546
7⁶√7±1.3830875543
8⁶√8±1.4142135624
9⁶√9±1.4422495703
10⁶√10±1.4677992676
11⁶√11±1.4913014754
12⁶√12±1.5130857494
13⁶√13±1.533406237
14⁶√14±1.5524632892
15⁶√15±1.5704178025
16⁶√16±1.587401052
17⁶√17±1.6035216215
18⁶√18±1.6188704069
19⁶√19±1.6335243031
20⁶√20±1.6475489724
21⁶√21±1.6610009562
22⁶√22±1.6739293087
23⁶√23±1.6863768795
24⁶√24±1.6983813296
25⁶√25±1.7099759467
26⁶√26±1.7211903057
27⁶√27±1.7320508076
28⁶√28±1.7425811235
29⁶√29±1.7528025632
30⁶√30±1.7627343833
31⁶√31±1.7723940455
32⁶√32±1.7817974363
33⁶√33±1.7909590531
34⁶√34±1.7998921638
35⁶√35±1.8086089434
36⁶√36±1.8171205928
37⁶√37±1.8254374412
38⁶√38±1.8335690352
39⁶√39±1.8415242173
40⁶√40±1.8493111943
41⁶√41±1.8569375973
42⁶√42±1.8644105355
43⁶√43±1.8717366429
44⁶√44±1.8789221206
45⁶√45±1.8859727741
46⁶√46±1.8928940464
47⁶√47±1.8996910486
48⁶√48±1.906368586
49⁶√49±1.9129311828
50⁶√50±1.9193831037
51⁶√51±1.9257283737
52⁶√52±1.9319707961
53⁶√53±1.9381139683
54⁶√54±1.9441612972
55⁶√55±1.9501160121
56⁶√56±1.9559811772
57⁶√57±1.9617597027
58⁶√58±1.9674543554
59⁶√59±1.9730677677
60⁶√60±1.9786024465
61⁶√61±1.9840607811
62⁶√62±1.9894450507
63⁶√63±1.9947574308
64⁶√64±2
65⁶√65±2.0051747452
66⁶√66±2.0102835672
67⁶√67±2.015328286
68⁶√68±2.0203106449
69⁶√69±2.025232315
70⁶√70±2.0300948992
71⁶√71±2.034899936
72⁶√72±2.0396489027
73⁶√73±2.0443432188
74⁶√74±2.0489842493
75⁶√75±2.0535733068
76⁶√76±2.058111655
77⁶√77±2.0626005103
78⁶√78±2.067041045
79⁶√79±2.0714343888
80⁶√80±2.0757816311
81⁶√81±2.0800838231
82⁶√82±2.0843419791
83⁶√83±2.0885570788
84⁶√84±2.0927300686
85⁶√85±2.0968618629
86⁶√86±2.1009533461
87⁶√87±2.1050053733
88⁶√88±2.1090187721
89⁶√89±2.1129943435
90⁶√90±2.116932863
91⁶√91±2.1208350821
92⁶√92±2.1247017286
93⁶√93±2.1285335083
94⁶√94±2.1323311056
95⁶√95±2.1360951841
96⁶√96±2.1398263879
97⁶√97±2.1435253421
98⁶√98±2.1471926537
99⁶√99±2.1508289121
100⁶√100±2.15443469

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.

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