Square Root Calculator

How to use the square root calculator

1. Choose square root, cube root or nth root. For nth root, enter the index n (4 for a fourth root, 5 for a fifth root and so on).
2. Type the number. Whole numbers, decimals (0.25), fractions (1/2) and negative numbers (−27) all work.

The answer comes as a decimal to 10 places and, where possible, in simplified radical form. You also see whether the number is a perfect square or cube, which two perfect squares it sits between, and its prime factorisation.

Square root formula and notation

The square root of x is the non-negative number that gives x when multiplied by itself:

√x = y where y × y = x and y ≥ 0

• The cube root ∛x is the number y with y³ = x.
• The nth root ⁿ√x is the number y with yⁿ = x.
• Roots are fractional powers: √x = x^(1/2), ∛x = x^(1/3) and ⁿ√x = x^(1/n). The exponent calculator handles powers like 8^(2/3).

Every positive number has two square roots, one positive and one negative (both 12 and −12 square to 144). The √ symbol means the positive one, called the principal square root.

How to simplify a square root

Simplest radical form means no perfect square is left under the root sign. The calculator does it with prime factorisation:

Step	√72	√180
Prime factors	72 = 2³ × 3²	180 = 2² × 3² × 5
Pull out pairs	(2² × 3²) × 2 = 36 × 2	(2² × 3²) × 5 = 36 × 5
Take roots	√36 × √2	√36 × √5
Simplified	6√2	6√5
Decimal	8.4852813742	13.416407865

For cube roots, pull out groups of three instead of pairs: 54 = 2 × 3³, so ∛54 = 3∛2. This is also how the quadratic formula calculator writes exact answers like 1 ± √2.

Finding a square root by hand

The Babylonian (or Heron’s) method improves a guess with each round:

1. Pick a guess g. For √10, g = 3 because 3² = 9 is close.
2. Divide: 10 ÷ 3 ≈ 3.3333.
3. Average: (3 + 3.3333) ÷ 2 ≈ 3.1667.
4. Repeat with the new guess: (3.1667 + 10 ÷ 3.1667) ÷ 2 ≈ 3.16228.

The true value is 3.1622776602, so two rounds already give five correct decimal places. Each round roughly doubles the number of correct digits.

Negative numbers and imaginary roots

A negative number has no real square root. In algebra and engineering it is written with the imaginary unit i, defined by i² = −1, so √(−72) = √72 × i = 6√2 i. Odd roots of negative numbers are ordinary real numbers: ∛(−27) = −3, because (−3)³ = −27. Even roots higher than two, such as the fourth root of −16, have no real value, and the calculator says so.

Perfect squares and perfect cubes from 1 to 20

n	n² (perfect square)	n³ (perfect cube)
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	729
10	100	1,000
11	121	1,331
12	144	1,728
13	169	2,197
14	196	2,744
15	225	3,375
16	256	4,096
17	289	4,913
18	324	5,832
19	361	6,859
20	400	8,000

Knowing these makes estimating easy. √72 must be between 8 and 9 because 72 sits between 64 and 81, and ∛100 is between 4 and 5 because 100 sits between 64 and 125. Square roots also turn up whenever you measure a diagonal, which is why the Pythagorean theorem calculator uses them for every answer.