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 = a1/6.

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.
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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 \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:

6th Root

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.

Calculate \sqrt[6]{x}

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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 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

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
0\sqrt[6]{0}0
1\sqrt[6]{1}±1
2\sqrt[6]{2}±1.1224620483
3\sqrt[6]{3}±1.2009369552
4\sqrt[6]{4}±1.2599210499
5\sqrt[6]{5}±1.307660486
6\sqrt[6]{6}±1.3480061546
7\sqrt[6]{7}±1.3830875543
8\sqrt[6]{8}±1.4142135624
9\sqrt[6]{9}±1.4422495703
10\sqrt[6]{10}±1.4677992676
11\sqrt[6]{11}±1.4913014754
12\sqrt[6]{12}±1.5130857494
13\sqrt[6]{13}±1.533406237
14\sqrt[6]{14}±1.5524632892
15\sqrt[6]{15}±1.5704178025
16\sqrt[6]{16}±1.587401052
17\sqrt[6]{17}±1.6035216215
18\sqrt[6]{18}±1.6188704069
19\sqrt[6]{19}±1.6335243031
20\sqrt[6]{20}±1.6475489724
21\sqrt[6]{21}±1.6610009562
22\sqrt[6]{22}±1.6739293087
23\sqrt[6]{23}±1.6863768795
24\sqrt[6]{24}±1.6983813296
25\sqrt[6]{25}±1.7099759467
26\sqrt[6]{26}±1.7211903057
27\sqrt[6]{27}±1.7320508076
28\sqrt[6]{28}±1.7425811235
29\sqrt[6]{29}±1.7528025632
30\sqrt[6]{30}±1.7627343833
31\sqrt[6]{31}±1.7723940455
32\sqrt[6]{32}±1.7817974363
33\sqrt[6]{33}±1.7909590531
34\sqrt[6]{34}±1.7998921638
35\sqrt[6]{35}±1.8086089434
36\sqrt[6]{36}±1.8171205928
37\sqrt[6]{37}±1.8254374412
38\sqrt[6]{38}±1.8335690352
39\sqrt[6]{39}±1.8415242173
40\sqrt[6]{40}±1.8493111943
41\sqrt[6]{41}±1.8569375973
42\sqrt[6]{42}±1.8644105355
43\sqrt[6]{43}±1.8717366429
44\sqrt[6]{44}±1.8789221206
45\sqrt[6]{45}±1.8859727741
46\sqrt[6]{46}±1.8928940464
47\sqrt[6]{47}±1.8996910486
48\sqrt[6]{48}±1.906368586
49\sqrt[6]{49}±1.9129311828
50\sqrt[6]{50}±1.9193831037
51\sqrt[6]{51}±1.9257283737
52\sqrt[6]{52}±1.9319707961
53\sqrt[6]{53}±1.9381139683
54\sqrt[6]{54}±1.9441612972
55\sqrt[6]{55}±1.9501160121
56\sqrt[6]{56}±1.9559811772
57\sqrt[6]{57}±1.9617597027
58\sqrt[6]{58}±1.9674543554
59\sqrt[6]{59}±1.9730677677
60\sqrt[6]{60}±1.9786024465
61\sqrt[6]{61}±1.9840607811
62\sqrt[6]{62}±1.9894450507
63\sqrt[6]{63}±1.9947574308
64\sqrt[6]{64}±2
65\sqrt[6]{65}±2.0051747452
66\sqrt[6]{66}±2.0102835672
67\sqrt[6]{67}±2.015328286
68\sqrt[6]{68}±2.0203106449
69\sqrt[6]{69}±2.025232315
70\sqrt[6]{70}±2.0300948992
71\sqrt[6]{71}±2.034899936
72\sqrt[6]{72}±2.0396489027
73\sqrt[6]{73}±2.0443432188
74\sqrt[6]{74}±2.0489842493
75\sqrt[6]{75}±2.0535733068
76\sqrt[6]{76}±2.058111655
77\sqrt[6]{77}±2.0626005103
78\sqrt[6]{78}±2.067041045
79\sqrt[6]{79}±2.0714343888
80\sqrt[6]{80}±2.0757816311
81\sqrt[6]{81}±2.0800838231
82\sqrt[6]{82}±2.0843419791
83\sqrt[6]{83}±2.0885570788
84\sqrt[6]{84}±2.0927300686
85\sqrt[6]{85}±2.0968618629
86\sqrt[6]{86}±2.1009533461
87\sqrt[6]{87}±2.1050053733
88\sqrt[6]{88}±2.1090187721
89\sqrt[6]{89}±2.1129943435
90\sqrt[6]{90}±2.116932863
91\sqrt[6]{91}±2.1208350821
92\sqrt[6]{92}±2.1247017286
93\sqrt[6]{93}±2.1285335083
94\sqrt[6]{94}±2.1323311056
95\sqrt[6]{95}±2.1360951841
96\sqrt[6]{96}±2.1398263879
97\sqrt[6]{97}±2.1435253421
98\sqrt[6]{98}±2.1471926537
99\sqrt[6]{99}±2.1508289121
100\sqrt[6]{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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