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- The square root of a number is the value that gives the number back when it is multiplied by itself, so √144 = 12 because 12 × 12 = 144.
- Every positive number has two square roots, one positive and one negative. This calculator shows the principal (positive) one.
- A perfect square such as 49 or 144 has a whole-number root. Most other numbers have roots that never end, so they are rounded or written as radicals like 6√2.
- Negative numbers have no real square root. They use the imaginary unit i, so √−16 = 4i.
How to use the square root calculator
Enter your number
Type any whole number or decimal into the Number box. The result updates as you type.
Read the square root
The main result shows the principal square root, rounded to six decimal places by default.
Check the extras
Underneath you will find the cube root, the square of your number, whether it is a perfect square or cube, and the simplified radical form.
Change the rounding
Open Rounding options to show anywhere from 2 to 10 decimal places.
How a square root is found
A square root undoes squaring. If b × b = a, then b is a square root of a. Because (−b) × (−b) gives the same answer, every positive number has two roots, a positive one and a negative one. The positive one is called the principal square root, and it is the one the √ symbol stands for.
Perfect squares such as 9, 25 and 144 have whole-number roots. Most other numbers, such as 2 or 72, have roots whose decimals continue forever without repeating. They are either rounded to a set number of decimals or written as a radical. A radical is simplified by pulling out the largest perfect-square factor. For 72 that factor is 36, which gives 6√2.
Square root rules
√a = b means b × b = a
√(a × b) = √a × √b
To simplify, pull out the largest perfect-square factor
√(−a) = √a × i
The last rule uses i, the imaginary unit, where i² = −1.
The cube root works the same way with three equal factors. The cube root of 27 is 3 because 3 × 3 × 3 = 27. Unlike square roots, cube roots of negative numbers are real: the cube root of −8 is −2. The calculator also shows the square of your number, which is the number multiplied by itself.
Worked examples
Example 1: a perfect square
Find the square root of 144.
Because 144 is a perfect square, the answer is a whole number.
Example 2: simplifying √72
Find the square root of 72 in simplest form.
The exact answer is 6√2. The decimal is a rounded copy of it.
Example 3: a negative number
Find the square root of −16.
There is no real square root, so the answer uses i. The cube root, on the other hand, is a real negative number.
Square roots of negative numbers
No real number multiplied by itself gives a negative result, because a positive times a positive and a negative times a negative are both positive. Mathematicians solve this by defining i as the square root of −1. Then √−16 = √16 × i = 4i. You will meet these imaginary numbers in algebra, electrical engineering and physics. If you only need real answers, treat a negative input as having no solution.
Common square roots
Square roots worth knowing
Values are rounded to six decimal places.
The perfect squares up to 144 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.
Where a square root calculator helps
Geometry
Find the side of a square from its area, or the diagonal of a rectangle with the Pythagorean theorem.
Algebra homework
Check answers and simplified radicals when you solve quadratic equations.
Building and DIY
Work out how long each side of a square room or plot is from its floor area.
Statistics
Standard deviation is the square root of the variance, so a quick root saves time.
If you know the area of a square, its side is the square root of the area. For other shapes, try the area calculator. To take a share of a result, use the percentage calculator, and for solids see the volume calculator.
Square root calculator FAQ
What is the square root of a number?
It is the value that gives the number back when multiplied by itself. The square root of 81 is 9 because 9 × 9 = 81.
How do I know if a number is a perfect square?
A whole number is a perfect square when its square root is also a whole number. The calculator checks this and shows Yes or No. The first perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
Can you take the square root of a negative number?
Not within the real numbers. The calculator gives the answer with the imaginary unit i instead, so the square root of −16 is 4i.
What does simplified radical form mean?
It means writing a root with the smallest possible number under the radical sign. The root of 72 becomes 6√2 because 72 = 36 × 2 and the square root of 36 is 6.
What is the difference between a square root and a cube root?
A square root multiplies two equal factors to reach the number, and a cube root multiplies three. The square root of 64 is 8 and the cube root of 64 is 4.
Why is the square root of 2 shown with decimals?
The square root of 2 is an irrational number, so its decimals never end or repeat. Any decimal form is a rounded value, while √2 is the exact one.
Related calculators
Need the smallest number that several numbers all divide into? Try the LCM calculator, which shows the least common multiple with steps.