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 5th root is usually denoted \sqrt[5]{x}, but it can also be written in exponential form with the base a and the exponent 1/5: x^{\frac{1}{5}}.

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 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:

\sqrt[5]{7776} = \sqrt[5]{6^{5}} = 6^\frac{5}{5} = {6^1} = 6

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:

5th Root

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 computes the fifth root of any non-negative real number.

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

Calculate \sqrt[5]{x}

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Next, we discuss the properties.

5th Root Properties

With a,b \in \mathbb{R^{+}} and k,m \in \mathbb{N}, the properties are as follows:

  • \sqrt[5]{x} = |a| = -a if a < 0 and a if ≥ 0
  • \sqrt[5]{ab}= \sqrt[5]{a} \sqrt[5]{b}
  • \sqrt[5]{a/b}= \frac{\sqrt[5]{a}}{\sqrt[5]{b}}
  • \sqrt[5]{0} = 0
  • \sqrt[5]{a^{-m}} = \frac{1}{\sqrt[5]a^{m}}
  • \sqrt[5k]{a^{m}} = \sqrt[5]{a^{km}}
  • \sqrt[5]{\sqrt[m]{a}} = \sqrt[5m]{a} = \sqrt[m]{\sqrt[5]{a}}

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:

  • \sqrt[5]{243} = 3
  • \sqrt[5]{32768} = \sqrt[5]{1024} \sqrt[5]{32} = 4 x 2 = 8
  • \sqrt[5]{100000/7776}= \frac{\sqrt[5]{100000}}{\sqrt[5]{7776}} = 10 / 6 = 5 / 4
  • \sqrt[5]{7^{-5}} = \frac{1}{\sqrt[5]{7^{5}}} = \frac{1}{7} = 1/7
  • \sqrt[10]{5^{4}} = \sqrt[5]{5^{2}} = \sqrt[5]{25}
  • \sqrt[5]{\sqrt[3]{a}} = \sqrt[5 x 3]{a} = \sqrt[15]{a}
  • \sqrt[5]{\sqrt[m]{7}} = \sqrt[6m]{7} = \sqrt[m]{\sqrt[5]{7}}

Frequently searched terms on this site include:

5th Root in Excel

In Excel you enter the syntax for x^{\frac{1}{5}} in a cell, like this:

=POWER(radicand,1/5)

For example, to calculate \sqrt[60]{64} 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.

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 5th root formula.

Next are frequently asked questions in the context of this article.

FAQs

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
0\sqrt[5]{0}0
1\sqrt[5]{1}1.1224620483
2\sqrt[5]{2}1.2009369552
3\sqrt[5]{3}1.2599210499
4\sqrt[5]{4}1.307660486
5\sqrt[5]{5}1.3480061546
6\sqrt[5]{6}1.3830875543
7\sqrt[5]{7}1.4142135624
8\sqrt[5]{8}1.4422495703
9\sqrt[5]{9}1.4677992676
10\sqrt[5]{10}1.4913014754
11\sqrt[5]{11}1.5130857494
12\sqrt[5]{12}1.533406237
13\sqrt[5]{13}1.5524632892
14\sqrt[5]{14}1.5704178025
15\sqrt[5]{15}1.587401052
16\sqrt[5]{16}1.6035216215
17\sqrt[5]{17}1.6188704069
18\sqrt[5]{18}1.6335243031
19\sqrt[5]{19}1.6475489724
20\sqrt[5]{20}1.6610009562
21\sqrt[5]{21}1.6739293087
22\sqrt[5]{22}1.6863768795
23\sqrt[5]{23}1.6983813296
24\sqrt[5]{24}1.7099759467
25\sqrt[5]{25}1.7211903057
26\sqrt[5]{26}1.7320508076
27\sqrt[5]{27}1.7425811235
28\sqrt[5]{28}1.7528025632
29\sqrt[5]{29}1.7627343833
30\sqrt[5]{30}1.7723940455
31\sqrt[5]{31}1.7817974363
32\sqrt[5]{32}1.7909590531
33\sqrt[5]{33}1.7998921638
34\sqrt[5]{34}1.8086089434
35\sqrt[5]{35}1.8171205928
36\sqrt[5]{36}1.8254374412
37\sqrt[5]{37}1.8335690352
38\sqrt[5]{38}1.8415242173
39\sqrt[5]{39}1.8493111943
40\sqrt[5]{40}1.8569375973
41\sqrt[5]{41}1.8644105355
42\sqrt[5]{42}1.8717366429
43\sqrt[5]{43}1.8789221206
44\sqrt[5]{44}1.8859727741
45\sqrt[5]{45}1.8928940464
46\sqrt[5]{46}1.8996910486
47\sqrt[5]{47}1.906368586
48\sqrt[5]{48}1.9129311828
49\sqrt[5]{49}1.9193831037
50\sqrt[5]{50}1.9257283737
51\sqrt[5]{51}1.9319707961
52\sqrt[5]{52}1.9381139683
53\sqrt[5]{53}1.9441612972
54\sqrt[5]{54}1.9501160121
55\sqrt[5]{55}1.9559811772
56\sqrt[5]{56}1.9617597027
57\sqrt[5]{57}1.9674543554
58\sqrt[5]{58}1.9730677677
59\sqrt[5]{59}1.9786024465
60\sqrt[5]{60}1.9840607811
61\sqrt[5]{61}1.9894450507
62\sqrt[5]{62}1.9947574308
63\sqrt[5]{63}2
64\sqrt[5]{64}2.0051747452
65\sqrt[5]{65}2.0102835672
66\sqrt[5]{66}2.015328286
67\sqrt[5]{67}2.0203106449
68\sqrt[5]{68}2.025232315
69\sqrt[5]{69}2.0300948992
70\sqrt[5]{70}2.034899936
71\sqrt[5]{71}2.0396489027
72\sqrt[5]{72}2.0443432188
73\sqrt[5]{73}2.0489842493
74\sqrt[5]{74}2.0535733068
75\sqrt[5]{75}2.058111655
76\sqrt[5]{76}2.0626005103
77\sqrt[5]{77}2.067041045
78\sqrt[5]{78}2.0714343888
79\sqrt[5]{79}2.0757816311
80\sqrt[5]{80}2.0800838231
81\sqrt[5]{81}2.0843419791
82\sqrt[5]{82}2.0885570788
83\sqrt[5]{83}2.0927300686
84\sqrt[5]{84}2.0968618629
85\sqrt[5]{85}2.1009533461
86\sqrt[5]{86}2.1050053733
87\sqrt[5]{87}2.1090187721
88\sqrt[5]{88}2.1129943435
89\sqrt[5]{89}2.116932863
90\sqrt[5]{90}2.1208350821
91\sqrt[5]{91}2.1247017286
92\sqrt[5]{92}2.1285335083
93\sqrt[5]{93}2.1323311056
94\sqrt[5]{94}2.1360951841
95\sqrt[5]{95}2.1398263879
96\sqrt[5]{96}2.1435253421
97\sqrt[5]{97}2.1471926537
98\sqrt[5]{98}2.1508289121
99\sqrt[5]{99}2.15443469
100\sqrt[5]{100}0

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 5th roots by using the search form in the header menu.

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