Your result will appear here
Fill in the fields and press Calculate.
Taking a root is the inverse of raising to a power: the answer to "what number squared is 144?" is √144 = 12. The square root is the most common, but a root can be of any degree — cube root, fourth root, and so on. Roots sit at the heart of many formulas from geometry to physics to finance, especially when moving from an area to a side, or from a volume to an edge.
Enter the number and the root degree, and the result is calculated instantly.
Root Calculator: step-by-step worked example
Simplify √72 and find its decimal value.
- Number
- 72
- Root degree
- 2
- 1
Split off the largest perfect-power factor
x = kⁿ × m (kⁿ = largest perfect power dividing x)
= 36 × 2
- 2
Take the perfect power out of the root
ⁿ√(kⁿ × m) = k × ⁿ√m
= 6√2
- 3
Decimal value
ⁿ√x = x^(1/n)
6 × 1.414214 = 8.4853
- 4
Check: raise the result to the power
(ⁿ√x)ⁿ = x
8.485281² = 72
- 8²64
- x = 7272
- 9²81
Good to know: Simplifying radicals: √72 = √(36 × 2) = 6√2. Even roots of negative numbers are not real numbers; odd roots keep the sign (∛−8 = −2).
Root Calculator — how it works
Definition
ⁿ√a is "the number whose nth power is a." The square root (n = 2) is usually written √ without the degree. So √144 = 12 because 12² = 144; ∛27 = 3 because 3³ = 27.
A root is the inverse of a power
An nth root equals the 1/n power: ⁿ√a = a^(1/n). That is why roots and exponents are two sides of the same coin — taking a root is taking a fractional exponent.
Watch out for
- Even root of a negative number: does not exist in the real numbers (√−4 is not real), because no real number squared is negative. An odd root (∛−8 = −2) is defined.
- Roots are often irrational: √2 = 1.41421… and most roots have infinite non-repeating decimals; the tool gives a decimal approximation.
- √ gives the positive root: even though x² = 9 has x = ±3, the √9 symbol denotes 3, the positive root.
Where it is used
- Geometry: from area to side (square area 144 → side √144 = 12), the Pythagorean theorem.
- Statistics: standard deviation is the square root of variance.
- Finance: the compound annual growth rate (CAGR) is an nth root.
- Physics: many formulas (period, speed) involve roots.
For the inverse operation use an exponent calculator.
Worked example
Take √144: we look for the number whose square is 144, and since 12 × 12 = 144, √144 = 12 — exactly the side of a square plot whose area is 144 m². For different degrees: ∛27 = 3 (the number whose cube is 27, the edge of a cube with volume 27), and ⁴√81 = 3 (the number whose fourth power is 81, since 3⁴ = 81). To see that a root is the inverse of a power: 3³ = 27 means ∛27 = 3; the operations undo each other.
Square and cube root table (1–30)
| n | √n | ∛n |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 1.4142 | 1.2599 |
| 3 | 1.7321 | 1.4422 |
| 4 | 2 | 1.5874 |
| 5 | 2.2361 | 1.71 |
| 6 | 2.4495 | 1.8171 |
| 7 | 2.6458 | 1.9129 |
| 8 | 2.8284 | 2 |
| 9 | 3 | 2.0801 |
| 10 | 3.1623 | 2.1544 |
| 11 | 3.3166 | 2.224 |
| 12 | 3.4641 | 2.2894 |
| 13 | 3.6056 | 2.3513 |
| 14 | 3.7417 | 2.4101 |
| 15 | 3.873 | 2.4662 |
| 16 | 4 | 2.5198 |
| 17 | 4.1231 | 2.5713 |
| 18 | 4.2426 | 2.6207 |
| 19 | 4.3589 | 2.6684 |
| 20 | 4.4721 | 2.7144 |
| 21 | 4.5826 | 2.7589 |
| 22 | 4.6904 | 2.802 |
| 23 | 4.7958 | 2.8439 |
| 24 | 4.899 | 2.8845 |
| 25 | 5 | 2.924 |
| 26 | 5.099 | 2.9625 |
| 27 | 5.1962 | 3 |
| 28 | 5.2915 | 3.0366 |
| 29 | 5.3852 | 3.0723 |
| 30 | 5.4772 | 3.1072 |
Frequently asked questions
How do I calculate a root?+
An nth root is the value whose nth power equals the number: √144 = 12 (12²=144), ∛27 = 3 (3³=27). Just enter the number and the root degree.
What is the difference between a square root and a cube root?+
A square root (degree 2) finds the value whose square is the number; a cube root (degree 3) finds the value whose cube is the number. √64 = 8 while ∛64 = 4.
Does a negative number have a root?+
An even root (like a square root) does not exist in the real numbers, since no real number squared is negative. But an odd root is defined: ∛−8 = −2.
How do roots relate to exponents?+
A root is a fractional exponent: ⁿ√a = a^(1/n). A square root is the 1/2 power, a cube root the 1/3 power. That makes roots and exponents inverse operations.
Why don't roots like √2 come out exact?+
Most roots are irrational — their decimal expansion is infinite and non-repeating (√2 = 1.41421356…). The tool shows a rounded approximation to a set number of places.
Why is a root used in standard deviation?+
Standard deviation is the square root of variance; the root brings the measure back to the original units of the data. To compute it directly use a standard deviation calculator.
About this page
- Publisher:
- Kalkoro — Teknolojig Bilişim Teknolojileri A.Ş., İstanbul
- Written by:
- Ahmet Tur — Editor (Math, date & time, unit conversion, physics & chemistry, everyday tools)
- Reviewed by:
- Kübra Yıldız — Editor
Parameters come from official sources; the last update date is shown at the bottom of the page.
Spotted a wrong result? Use the "Report a bug" button at the top of the page; every report is reviewed.
How we calculate and our accuracy policy →
Results are for information only. For official matters the relevant authority's calculation prevails.
Last updated: