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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 4-th root is usually denoted 4√x, but it can also be written in exponential form with the base x and the exponent 1/4: x^1/4 or x1/4.
Read on to learn everything about these numbers, including the properties, and make sure to check out our calculator.
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:
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.
If the index n of a root is is even, such as in the case here with n = 4, then every positive number
The positive is also known as principal 4th root
and the negative 4th root
Together, they are written as
Let’s have a look the previous example:
Proof:
-74 = 2401
734 = 2401
Both values multiplied 4 x by itself = 2401.
Make sure to understand that multiplying the 4th root four times by itself produces the original number (not the 4th root x 4).
The positive root is always the principal.
This should sound familiar if you have read our square root article – the home page.
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:
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 equation sign and called the root.
4th Root Symbol
The symbol √ is called radical sign, or radix.
Ahead is a few words regarding our calculator.
About our 4th Root Calculator
Our calculator at the top of this page 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.
Note that our tool works both ways; that is the math is bidirectional.
Use the up and down arrows (called spinners) to increase or decrease the input value.
You can change both, the upper as well as the lower input field of our 4-th root calculator.
If this online app has been of use to you bookmark it now.
Next, we discuss the properties.
4-th Root Properties
With
= -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:
4th 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:
4th Root in Excel
In Excel you enter the syntax for
in a cell, like this:
=POWER(radicand,1/4)
For example, to calculate
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.
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.
If something about fourth roots needs clarification do not hesitate getting in touch with us.
We are constantly aiming to improve this site, and truly appreciate your feedback!
Websites with calculators which are related to the root of a number can be found in the “recommended sites” section” in the sidebar.
Table of Fourth Roots
| Radicand | Symbol | 4th Root |
|---|---|---|
| 0 | ⁴√0 | 0 |
| 1 | ⁴√1 | ±1 |
| 2 | ⁴√2 | ±1.189207115 |
| 3 | ⁴√3 | ±1.316074013 |
| 4 | ⁴√4 | ±1.4142135624 |
| 5 | ⁴√5 | ±1.4953487812 |
| 6 | ⁴√6 | ±1.5650845801 |
| 7 | ⁴√7 | ±1.6265765617 |
| 8 | ⁴√8 | ±1.6817928305 |
| 9 | ⁴√9 | ±1.7320508076 |
| 10 | ⁴√10 | ±1.77827941 |
| 11 | ⁴√11 | ±1.8211602868 |
| 12 | ⁴√12 | ±1.8612097182 |
| 13 | ⁴√13 | ±1.8988289221 |
| 14 | ⁴√14 | ±1.9343364203 |
| 15 | ⁴√15 | ±1.9679896713 |
| 16 | ⁴√16 | ±2 |
| 17 | ⁴√17 | ±2.0305431849 |
| 18 | ⁴√18 | ±2.0597671439 |
| 19 | ⁴√19 | ±2.0877976299 |
| 20 | ⁴√20 | ±2.1147425269 |
| 21 | ⁴√21 | ±2.1406951429 |
| 22 | ⁴√22 | ±2.1657367707 |
| 23 | ⁴√23 | ±2.1899387031 |
| 24 | ⁴√24 | ±2.2133638394 |
| 25 | ⁴√25 | ±2.2360679775 |
| 26 | ⁴√26 | ±2.2581008644 |
| 27 | ⁴√27 | ±2.279507057 |
| 28 | ⁴√28 | ±2.3003266338 |
| 29 | ⁴√29 | ±2.3205957871 |
| 30 | ⁴√30 | ±2.3403473193 |
| 31 | ⁴√31 | ±2.3596110618 |
| 32 | ⁴√32 | ±2.37841423 |
| 33 | ⁴√33 | ±2.3967817269 |
| 34 | ⁴√34 | ±2.4147364028 |
| 35 | ⁴√35 | ±2.4322992791 |
| 36 | ⁴√36 | ±2.4494897428 |
| 37 | ⁴√37 | ±2.4663257146 |
| 38 | ⁴√38 | ±2.4828237962 |
| 39 | ⁴√39 | ±2.4989993994 |
| 40 | ⁴√40 | ±2.5148668594 |
| 41 | ⁴√41 | ±2.5304395344 |
| 42 | ⁴√42 | ±2.545729895 |
| 43 | ⁴√43 | ±2.560749602 |
| 44 | ⁴√44 | ±2.5755095769 |
| 45 | ⁴√45 | ±2.5900200641 |
| 46 | ⁴√46 | ±2.6042906871 |
| 47 | ⁴√47 | ±2.6183304987 |
| 48 | ⁴√48 | ±2.6321480259 |
| 49 | ⁴√49 | ±2.6457513111 |
| 50 | ⁴√50 | ±2.6591479485 |
| 51 | ⁴√51 | ±2.6723451178 |
| 52 | ⁴√52 | ±2.6853496143 |
| 53 | ⁴√53 | ±2.6981678764 |
| 54 | ⁴√54 | ±2.7108060108 |
| 55 | ⁴√55 | ±2.7232698153 |
| 56 | ⁴√56 | ±2.7355647997 |
| 57 | ⁴√57 | ±2.7476962051 |
| 58 | ⁴√58 | ±2.7596690211 |
| 59 | ⁴√59 | ±2.7714880025 |
| 60 | ⁴√60 | ±2.7831576837 |
| 61 | ⁴√61 | ±2.7946823927 |
| 62 | ⁴√62 | ±2.8060662633 |
| 63 | ⁴√63 | ±2.8173132473 |
| 64 | ⁴√64 | ±2.8284271247 |
| 65 | ⁴√65 | ±2.8394115144 |
| 66 | ⁴√66 | ±2.8502698828 |
| 67 | ⁴√67 | ±2.8610055526 |
| 68 | ⁴√68 | ±2.871621711 |
| 69 | ⁴√69 | ±2.8821214171 |
| 70 | ⁴√70 | ±2.8925076085 |
| 71 | ⁴√71 | ±2.9027831082 |
| 72 | ⁴√72 | ±2.9129506302 |
| 73 | ⁴√73 | ±2.9230127857 |
| 74 | ⁴√74 | ±2.9329720877 |
| 75 | ⁴√75 | ±2.9428309564 |
| 76 | ⁴√76 | ±2.9525917237 |
| 77 | ⁴√77 | ±2.9622566377 |
| 78 | ⁴√78 | ±2.9718278662 |
| 79 | ⁴√79 | ±2.9813075013 |
| 80 | ⁴√80 | ±2.9906975624 |
| 81 | ⁴√81 | ±3 |
| 82 | ⁴√82 | ±3.0092166984 |
| 83 | ⁴√83 | ±3.0183494793 |
| 84 | ⁴√84 | ±3.027400104 |
| 85 | ⁴√85 | ±3.0363702767 |
| 86 | ⁴√86 | ±3.0452616465 |
| 87 | ⁴√87 | ±3.05407581 |
| 88 | ⁴√88 | ±3.0628143136 |
| 89 | ⁴√89 | ±3.0714786556 |
| 90 | ⁴√90 | ±3.0800702882 |
| 91 | ⁴√91 | ±3.0885906194 |
| 92 | ⁴√92 | ±3.0970410147 |
| 93 | ⁴√93 | ±3.1054227991 |
| 94 | ⁴√94 | ±3.1137372585 |
| 95 | ⁴√95 | ±3.1219856414 |
| 96 | ⁴√96 | ±3.1301691601 |
| 97 | ⁴√97 | ±3.1382889927 |
| 98 | ⁴√98 | ±3.1463462836 |
| 99 | ⁴√99 | ±3.1543421455 |
| 100 | ⁴√100 | ±3.1622776602 |
Ahead is the wrap up of our information regarding the 4th root of a number.
4th Root Summary
You have made it to the end of our article about the 4 th root.
The 4th root b of a number a is such that b4 = a; b = ±⁴√a.
⁴√ is the 4-th root symbol consisting of the radix and the index 4.
Note that you can find the fourth root of many numbers by using the search form in the header menu.
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– Article written by Mark, last updated on November 26th, 2023