What is a square root?

The square root of a number x is the value y such that y² = x. Written as √x or x^(1/2). √9 = 3 (because 3²=9). √2 ≈ 1.41421. √16 = 4. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Irrational numbers like √2, √3, √5 have non-repeating, non-terminating decimal expansions.

The cube root of x is the value y such that y³ = x. Written as ∛x or x^(1/3). ∛8 = 2 (because 2³=8). ∛27 = 3. ∛64 = 4. ∛125 = 5. Cube roots of non-perfect cubes are irrational: ∛2 ≈ 1.2599. Unlike square roots, cube roots of negative numbers are real: ∛(−8) = −2.

How do I estimate a square root without a calculator?

Find the nearest perfect squares above and below. To estimate √50: √49=7, √64=8. So √50 is between 7 and 8, closer to 7. More precisely: 50−49=1; 64−49=15. Fraction = 1/15 ≈ 0.067. Estimate ≈ 7.07. Actual: 7.071. Another method: Newton's method — guess g, refine as (g + x/g)/2 repeatedly.

Root Calculator

Calculate the square root (√), cube root, or any nᵗʰ root. n = 2 for square root, n = 3 for cube root, etc.

Root Calculator — Guide

Root	n	Notation	Example
Square root	2	√x	√9 = 3
Cube root	3	∛x	∛27 = 3
4th root	4	∜x	∜256 = 4
nth root	n	x^(1/n)	64^(1/6) = 2

Formula

ⁿ√x = x^(1/n)

Related Calculators

Types of Roots and When to Use Them

The square root (√) is the most common, used in geometry (Pythagorean theorem), physics (RMS values), and statistics (standard deviation). The cube root (∛) appears in volume problems — if a cube has volume 125 cm³, its side length is ∛125 = 5 cm. The nth root generalises this: the 4th root of 16 is 2 because 2⁴ = 16. Roots are the inverse of powers, so understanding one means understanding the other.

In finance, the compound annual growth rate (CAGR) formula uses the nth root: CAGR = (End Value / Start Value)^(1/n) − 1, where n is the number of years. A $1,000 investment growing to $1,610 over 5 years has a CAGR of ⁵√(1.61) − 1 ≈ 10% per year.

Root Reference Table

Expression	Value	Note
√144	12	Perfect square
√2	1.4142…	Diagonal of unit square
∛27	3	Side of cube, volume 27
⁴√81	3	3⁴ = 81
√-1	i	Imaginary unit