4th Root

In math, the 4th root b of a number a is such that b4 = a. By definition, when you multiply b by itself 4 times you get the value of a.

A 4th root is usually denoted \sqrt[4]{x}, but it can also be written in exponential form with the base a and the exponent 1/4: x^{\frac{1}{4}}.

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 4th root of a number as the inverse operation to elevating a number to the power of 4.

Definition

Whereas in exponentiation elevating a number a to the power of 4 is defined as a4 = b, the 4th root b is defined as b = a1/4.

For example with a = 2401 we get:

\sqrt[4]{2401} = \sqrt[4]{7^{4}} = 7^\frac{4}{4} = {7^1} = 7

In other words, the 4th root of 2401 is 7, because 7 times 7 times 7 times 7 is 7.

Keep reading to learn an important property of the fourth root.
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If the index n of a root is is even, such as in the case here with n = 4, then every positive number \in \mathbb{R} has two such roots:

The positive is also known as principal 4th root \sqrt[n]{x}, and the negative 4th root -\sqrt[n]{x}.

Together, they are written as \pm\sqrt[4]{x}.

Let’s have a look the previous example:

\sqrt[4]{2401} = \sqrt[4]{7^{4}} = 7^\frac{4}{4} = {7^1} = \pm 7

Proof:

-74 = 2401

734 = 2401

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 fourth root are:

4th Root

The radix sign, which tells us that it is a mathematical root, and the index of 4, which tells us that it is the 4-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.

4th Root Symbol

The symbol √ is called radical sign, or radix.

Ahead is our calculator.

4th Root Calculator

Our calculator computes the fourth 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[4]{x}

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Next, we discuss the properties.

4th Root Properties

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

  • \sqrt[4]{x} = |a| = -a if a < 0 and a if ≥ 0
  • \sqrt[4]{ab}= \sqrt[4]{a} \sqrt[4]{b}
  • \sqrt[4]{a/b}= \frac{\sqrt[4]{a}}{\sqrt[4]{b}}
  • \sqrt[4]{0} = 0
  • \sqrt[4]{a^{-m}} = \frac{1}{\sqrt[4]a^{m}}
  • \sqrt[4k]{a^{m}} = \sqrt[4]{a^{km}}
  • \sqrt[4]{\sqrt[m]{a}} = \sqrt[4m]{a} = \sqrt[m]{\sqrt[4]{a}}

The most important property is the first; the negative number tends to be forgotten. Read on to see the examples:

4th Root Examples

We use the list of properties above to show you some examples in the order of appearance:

  • \sqrt[4]{4096} = \pm 8
  • \sqrt[4]{20736} = \sqrt[4]{81} \sqrt[4]{256} = 3 x 4 = \pm 12
  • \sqrt[4]{256/16}= \frac{\sqrt[4]{256}}{\sqrt[4]{16}} = 16/2 = \pm 8
  • \sqrt[4]{5^{-4}} = \frac{1}{\sqrt[4]{5^{4}}} = \frac{1}{5} = \pm 1/5
  • \sqrt[4]{9^{4}} = \sqrt{9^{2}} = \sqrt{81} = \pm 9
  • \sqrt[4]{\sqrt[3]{125}} = \sqrt[4 x 3]{125} = \pm \sqrt[12]{125}
  • \sqrt[4]{\sqrt[m]{6}} = \sqrt[4m]{5} = \sqrt[m]{\sqrt[4]{6}}

Frequently searched terms on this site include:

4th Root in Excel

In Excel you enter the syntax for x^{\frac{1}{4}} in a cell, like this:

=POWER(radicand,1/4)

For example, to calculate \sqrt[4]{64} insert =POWER(64,1/4).

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

How to Calculate the 4th Root

A very efficient procedure for extracting the fourth 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 4th root formula.

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

FAQs

What is the 4th Root Called?

The 4th root is called fourth root.

How Do You Type a 4th Root on a Computer?

On a Windows computer hit the Alt key and type 8732 simultaneously, on a Mac hit Option and type 221C.

How Do You do 4th Roots on a Calculator?

Look for the power function, then insert the 4th root as x^(1/4).

Table of Fourth Roots

RadicandSymbol4th Root
0\sqrt[4]{0}0
1\sqrt[4]{1}±1
2\sqrt[4]{2}±1.189207115
3\sqrt[4]{3}±1.316074013
4\sqrt[4]{4}±1.4142135624
5\sqrt[4]{5}±1.4953487812
6\sqrt[4]{6}±1.5650845801
7\sqrt[4]{7}±1.6265765617
8\sqrt[4]{8}±1.6817928305
9\sqrt[4]{9}±1.7320508076
10\sqrt[4]{10}±1.77827941
11\sqrt[4]{11}±1.8211602868
12\sqrt[4]{12}±1.8612097182
13\sqrt[4]{13}±1.8988289221
14\sqrt[4]{14}±1.9343364203
15\sqrt[4]{15}±1.9679896713
16\sqrt[4]{16}±2
17\sqrt[4]{17}±2.0305431849
18\sqrt[4]{18}±2.0597671439
19\sqrt[4]{19}±2.0877976299
20\sqrt[4]{20}±2.1147425269
21\sqrt[4]{21}±2.1406951429
22\sqrt[4]{22}±2.1657367707
23\sqrt[4]{23}±2.1899387031
24\sqrt[4]{24}±2.2133638394
25\sqrt[4]{25}±2.2360679775
26\sqrt[4]{26}±2.2581008644
27\sqrt[4]{27}±2.279507057
28\sqrt[4]{28}±2.3003266338
29\sqrt[4]{29}±2.3205957871
30\sqrt[4]{30}±2.3403473193
31\sqrt[4]{31}±2.3596110618
32\sqrt[4]{32}±2.37841423
33\sqrt[4]{33}±2.3967817269
34\sqrt[4]{34}±2.4147364028
35\sqrt[4]{35}±2.4322992791
36\sqrt[4]{36}±2.4494897428
37\sqrt[4]{37}±2.4663257146
38\sqrt[4]{38}±2.4828237962
39\sqrt[4]{39}±2.4989993994
40\sqrt[4]{40}±2.5148668594
41\sqrt[4]{41}±2.5304395344
42\sqrt[4]{42}±2.545729895
43\sqrt[4]{43}±2.560749602
44\sqrt[4]{44}±2.5755095769
45\sqrt[4]{45}±2.5900200641
46\sqrt[4]{46}±2.6042906871
47\sqrt[4]{47}±2.6183304987
48\sqrt[4]{48}±2.6321480259
49\sqrt[4]{49}±2.6457513111
50\sqrt[4]{50}±2.6591479485
51\sqrt[4]{51}±2.6723451178
52\sqrt[4]{52}±2.6853496143
53\sqrt[4]{53}±2.6981678764
54\sqrt[4]{54}±2.7108060108
55\sqrt[4]{55}±2.7232698153
56\sqrt[4]{56}±2.7355647997
57\sqrt[4]{57}±2.7476962051
58\sqrt[4]{58}±2.7596690211
59\sqrt[4]{59}±2.7714880025
60\sqrt[4]{60}±2.7831576837
61\sqrt[4]{61}±2.7946823927
62\sqrt[4]{62}±2.8060662633
63\sqrt[4]{63}±2.8173132473
64\sqrt[4]{64}±2.8284271247
65\sqrt[4]{65}±2.8394115144
66\sqrt[4]{66}±2.8502698828
67\sqrt[4]{67}±2.8610055526
68\sqrt[4]{68}±2.871621711
69\sqrt[4]{69}±2.8821214171
70\sqrt[4]{70}±2.8925076085
71\sqrt[4]{71}±2.9027831082
72\sqrt[4]{72}±2.9129506302
73\sqrt[4]{73}±2.9230127857
74\sqrt[4]{74}±2.9329720877
75\sqrt[4]{75}±2.9428309564
76\sqrt[4]{76}±2.9525917237
77\sqrt[4]{77}±2.9622566377
78\sqrt[4]{78}±2.9718278662
79\sqrt[4]{79}±2.9813075013
80\sqrt[4]{80}±2.9906975624
81\sqrt[4]{81}±3
82\sqrt[4]{82}±3.0092166984
83\sqrt[4]{83}±3.0183494793
84\sqrt[4]{84}±3.027400104
85\sqrt[4]{85}±3.0363702767
86\sqrt[4]{86}±3.0452616465
87\sqrt[4]{87}±3.05407581
88\sqrt[4]{88}±3.0628143136
89\sqrt[4]{89}±3.0714786556
90\sqrt[4]{90}±3.0800702882
91\sqrt[4]{91}±3.0885906194
92\sqrt[4]{92}±3.0970410147
93\sqrt[4]{93}±3.1054227991
94\sqrt[4]{94}±3.1137372585
95\sqrt[4]{95}±3.1219856414
96\sqrt[4]{96}±3.1301691601
97\sqrt[4]{97}±3.1382889927
98\sqrt[4]{98}±3.1463462836
99\sqrt[4]{99}±3.1543421455
100\sqrt[4]{100}±3.1622776602

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 4th roots by using the search form in the header menu.

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