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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.
How is it calculated?
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.
FAQ
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.