In math, the **5th root** *b* of a number *a* is such that b^{5} = 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 a^{5} = b, the 5th root *b* is defined as b = a^{1/5}.

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.

On this Site You Can Find all About 5th Roots, Including a Calculator! Click To Tweet

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 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}

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

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

Radicand | Symbol | 5th 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

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