5th Root

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In math, the 5th root b of a number a is such that b⁵ = a. By definition, when you multiply b by itself 5 times you get the value of a.

A 5-th root is usually denoted ⁵√x, but it can also be written in exponential form with the base x and the exponent 1/5: x^1/5 or x^1/5.

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 5th root of a number as the inverse operation to elevating a number to the power of 5.

Definition

Whereas in exponentiation elevating a number a to the power of 5 is defined as a⁵ = b, the 5th root b is defined as b = a^1/5.

For example with a = 7776 we get:

In other words, the 5th root of 7776 is 6, because 6 times 6 times 6 times 6 times 6 is 7776.

As the index n (5) is odd, 5 is the only real 5th root.

You may be interested what happens if the nth Root index n of a root is is even.

Keep reading to learn everything about the fifth root.

Next, we explain how the parts are called. Keep reading to learn all about the topic.

Parts

As depicted, the parts of the fifth root are:

The radix sign, which tells us that it is a mathematical root, and the index of 5, which tells us that it is the 5-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.

5th Root Symbol

The symbol √ is called radical sign, or radix.

Ahead is our calculator.

About our 5th Root Calculator

Our calculator at the top of this page computes the fifth root of any non-negative real number.

Just enter a valid radicand; you then automatically obtain the result.

If this app has been of use to you bookmark it now.

Next, we discuss the properties.

5th 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:

5th 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:

5th Root in Excel

In Excel you enter the syntax for

in a cell, like this:

=POWER(radicand,1/5)

For example, to calculate

insert =POWER(60,1/6).

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

How to Calculate the 5th Root

A very efficient procedure for extracting the fifth 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 5th root formula.

Table of Fifth Roots

Radicand	Symbol	5th Root
0	⁵√0	0
1	⁵√1	1
2	⁵√2	1.148698355
3	⁵√3	1.2457309396
4	⁵√4	1.3195079108
5	⁵√5	1.3797296615
6	⁵√6	1.4309690811
7	⁵√7	1.4757731616
8	⁵√8	1.5157165665
9	⁵√9	1.5518455739
10	⁵√10	1.5848931925
11	⁵√11	1.6153942662
12	⁵√12	1.6437518295
13	⁵√13	1.6702776523
14	⁵√14	1.6952182031
15	⁵√15	1.7187719276
16	⁵√16	1.7411011266
17	⁵√17	1.7623403478
18	⁵√18	1.782602458
19	⁵√19	1.8019831273
20	⁵√20	1.820564203
21	⁵√21	1.8384162873
22	⁵√22	1.8556007363
23	⁵√23	1.8721712306
24	⁵√24	1.8881750226
25	⁵√25	1.9036539387
26	⁵√26	1.9186451916
27	⁵√27	1.9331820449
28	⁵√28	1.9472943612
29	⁵√29	1.9610090575
30	⁵√30	1.9743504858
31	⁵√31	1.9873407547
32	⁵√32	2
33	⁵√33	2.0123466171
34	⁵√34	2.0243974585
35	⁵√35	2.0361680046
36	⁵√36	2.0476725111
37	⁵√37	2.0589241365
38	⁵√38	2.0699350541
39	⁵√39	2.0807165493
40	⁵√40	2.0912791052
41	⁵√41	2.1016324783
42	⁵√42	2.111785765
43	⁵√43	2.1217474608
44	⁵√44	2.1315255133
45	⁵√45	2.1411273683
46	⁵√46	2.1505600128
47	⁵√47	2.1598300118
48	⁵√48	2.1689435424
49	⁵√49	2.1779064245
50	⁵√50	2.1867241479
51	⁵√51	2.1954018974
52	⁵√52	2.2039445754
53	⁵√53	2.2123568223
54	⁵√54	2.2206430349
55	⁵√55	2.228807384
56	⁵√56	2.2368538294
57	⁵√57	2.2447861344
58	⁵√58	2.2526078784
59	⁵√59	2.2603224696
60	⁵√60	2.2679331553
61	⁵√61	2.2754430321
62	⁵√62	2.2828550557
63	⁵√63	2.2901720489
64	⁵√64	2.29739671
65	⁵√65	2.3045316198
66	⁵√66	2.3115792487
67	⁵√67	2.318541963
68	⁵√68	2.3254220304
69	⁵√69	2.3322216262
70	⁵√70	2.3389428374
71	⁵√71	2.3455876685
72	⁵√72	2.352158045
73	⁵√73	2.3586558182
74	⁵√74	2.3650827686
75	⁵√75	2.3714406098
76	⁵√76	2.3777309916
77	⁵√77	2.3839555035
78	⁵√78	2.3901156774
79	⁵√79	2.3962129905
80	⁵√80	2.402248868
81	⁵√81	2.4082246853
82	⁵√82	2.4141417706
83	⁵√83	2.420001407
84	⁵√84	2.4258048343
85	⁵√85	2.4315532515
86	⁵√86	2.437247818
87	⁵√87	2.4428896557
88	⁵√88	2.4484798507
89	⁵√89	2.4540194545
90	⁵√90	2.4595094858
91	⁵√91	2.4649509317
92	⁵√92	2.470344749
93	⁵√93	2.4756918656
94	⁵√94	2.4809931816
95	⁵√95	2.4862495702
96	⁵√96	2.4914618792
97	⁵√97	2.4966309317
98	⁵√98	2.5017575271
99	⁵√99	2.5068424421
100	⁵√100	2.5118864315

Ahead is the bottom line.

Conclusion

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Note that you can find the value of many 5th roots by using the search form in the header menu.

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– Article written by Mark, last updated on November 26th, 2023