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5th Root


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 x1/5.

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 = a1/5.

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.
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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.

5th Root Calculator

Our calculator above 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.

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 5th Root Called?

The 5th root is called fifth root.

How Do You Type a 5th Root on a Computer?

Write the 5 as ⁵ followed by the radix sign √: ALT + 8309 + ALT 251.

How Do You do 5th Roots on a Calculator?

Look for the power function, then insert the 5th root as x^(1/5).



Table of Fifth Roots







RadicandSymbol5th Root
0⁵√00
1⁵√11
2⁵√21.148698355
3⁵√31.2457309396
4⁵√41.3195079108
5⁵√51.3797296615
6⁵√61.4309690811
7⁵√71.4757731616
8⁵√81.5157165665
9⁵√91.5518455739
10⁵√101.5848931925
11⁵√111.6153942662
12⁵√121.6437518295
13⁵√131.6702776523
14⁵√141.6952182031
15⁵√151.7187719276
16⁵√161.7411011266
17⁵√171.7623403478
18⁵√181.782602458
19⁵√191.8019831273
20⁵√201.820564203
21⁵√211.8384162873
22⁵√221.8556007363
23⁵√231.8721712306
24⁵√241.8881750226
25⁵√251.9036539387
26⁵√261.9186451916
27⁵√271.9331820449
28⁵√281.9472943612
29⁵√291.9610090575
30⁵√301.9743504858
31⁵√311.9873407547
32⁵√322
33⁵√332.0123466171
34⁵√342.0243974585
35⁵√352.0361680046
36⁵√362.0476725111
37⁵√372.0589241365
38⁵√382.0699350541
39⁵√392.0807165493
40⁵√402.0912791052
41⁵√412.1016324783
42⁵√422.111785765
43⁵√432.1217474608
44⁵√442.1315255133
45⁵√452.1411273683
46⁵√462.1505600128
47⁵√472.1598300118
48⁵√482.1689435424
49⁵√492.1779064245
50⁵√502.1867241479
51⁵√512.1954018974
52⁵√522.2039445754
53⁵√532.2123568223
54⁵√542.2206430349
55⁵√552.228807384
56⁵√562.2368538294
57⁵√572.2447861344
58⁵√582.2526078784
59⁵√592.2603224696
60⁵√602.2679331553
61⁵√612.2754430321
62⁵√622.2828550557
63⁵√632.2901720489
64⁵√642.29739671
65⁵√652.3045316198
66⁵√662.3115792487
67⁵√672.318541963
68⁵√682.3254220304
69⁵√692.3322216262
70⁵√702.3389428374
71⁵√712.3455876685
72⁵√722.352158045
73⁵√732.3586558182
74⁵√742.3650827686
75⁵√752.3714406098
76⁵√762.3777309916
77⁵√772.3839555035
78⁵√782.3901156774
79⁵√792.3962129905
80⁵√802.402248868
81⁵√812.4082246853
82⁵√822.4141417706
83⁵√832.420001407
84⁵√842.4258048343
85⁵√852.4315532515
86⁵√862.437247818
87⁵√872.4428896557
88⁵√882.4484798507
89⁵√892.4540194545
90⁵√902.4595094858
91⁵√912.4649509317
92⁵√922.470344749
93⁵√932.4756918656
94⁵√942.4809931816
95⁵√952.4862495702
96⁵√962.4914618792
97⁵√972.4966309317
98⁵√982.5017575271
99⁵√992.5068424421
100⁵√1002.5118864315

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Conclusion

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