Simplest radical form
Get exact answers such as √72 = 6√2 or ∛108 = 3∛4, not just a long decimal.
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Square Root Calculator
Find the square root of any number and simplify it to exact radical form, like √72 = 6√2. This square root calculator also handles cube roots and nth roots, fractions, decimals and negative numbers, and shows every step so you can do it yourself next time.
- Simplest radical form
- Up to 20 decimal places
- Cube & nth roots
- Step-by-step working
Square root calculator
Try:
√72 = 6√2 ≈ 8.485281
Decimal value8.485281
Simplest form6√2
Perfect square?No
Exponent form721/2
Simplify √72 step by step
Prime factors are grouped into complete sets. Each set moves outside the root, and leftovers stay inside.
Where the root sits
Find it by hand: Babylonian and Newton method
| Step | Guess | Guess raised to n | Error |
|---|
Reference tables
Perfect Squares, Cubes and Square Roots Chart
Switch tables below. If your number appears, its row is highlighted.
| n | n² |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| n | n² |
|---|---|
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
| 16 | 256 |
| 17 | 289 |
| 18 | 324 |
| 19 | 361 |
| 20 | 400 |
| n | n² |
|---|---|
| 21 | 441 |
| 22 | 484 |
| 23 | 529 |
| 24 | 576 |
| 25 | 625 |
| 26 | 676 |
| 27 | 729 |
| 28 | 784 |
| 29 | 841 |
| 30 | 900 |
| n | n² |
|---|---|
| 31 | 961 |
| 32 | 1,024 |
| 33 | 1,089 |
| 34 | 1,156 |
| 35 | 1,225 |
| 36 | 1,296 |
| 37 | 1,369 |
| 38 | 1,444 |
| 39 | 1,521 |
| 40 | 1,600 |
| n | n² |
|---|---|
| 41 | 1,681 |
| 42 | 1,764 |
| 43 | 1,849 |
| 44 | 1,936 |
| 45 | 2,025 |
| 46 | 2,116 |
| 47 | 2,209 |
| 48 | 2,304 |
| 49 | 2,401 |
| 50 | 2,500 |
Why use this one
Why This Square Root Calculator Stands Out
Most root tools stop at a rounded decimal. This one gives you the exact answer, the simplified form and the reasoning behind it.
Up to 20 exact decimals
Big-number arithmetic keeps every digit correct, even for huge inputs where normal calculators round off.
Visual factor grouping
Colored chips show which prime factors leave the root and which stay inside, so the method clicks.
Any root from 2 to 10
Switch between square, cube and higher roots without changing tools.
Fractions, decimals, negatives
Enter 2/3, 0.25 or −16. The calculator rationalizes fractions and uses i for imaginary roots.
Learn to do it by hand
A number line and a Babylonian method table show how to estimate and refine a root yourself.
Quick start
How to Use the Square Root Calculator
Type a number
Enter a whole number, a decimal, a fraction like 5/8, or a negative number.
Choose the root
Keep the square root, or pick a cube root or any root up to the 10th.
Set the decimals
Choose how many decimal places you need, from 0 up to 20.
Read the steps
See the simplified form, the decimal, a check, and a full explanation beside the result.
The basics
What Is a Square Root?
The square root of a number is the value that gives that number when you multiply it by itself. For example, the square root of 49 is 7, because 7 × 7 = 49.
√x = y means y² = x
Every positive number actually has two square roots: one positive and one negative. Both 7 and −7 square to 49. However, the symbol √ always means the positive one, called the principal square root. That is the value this calculator shows first.
Numbers such as 1, 4, 9, 16 and 25 are perfect squares, so their roots are whole numbers. Most other roots, like √2 or √10, are irrational. Their decimals never end or repeat, which is why the exact radical form is so useful. You can read more in this overview of square roots.
Three methods
How to Find a Square Root by Hand
1. Prime factorization
Break the number into prime factors and pair them up. For instance, 144 = 2⁴ × 3², so √144 = 2² × 3 = 12. This method gives exact answers and also shows how to simplify roots that are not whole numbers.
2. Estimate with perfect squares
Find the perfect squares on either side of your number. Since 64 < 72 < 81, you know that √72 is between 8 and 9. Because 72 sits a little under halfway, a good first guess is about 8.5.
3. The Babylonian method
Start with a guess, then average it with the number divided by that guess. Repeat until the answer stops changing. For √10, start at 3:
- (3 + 10 ÷ 3) ÷ 2 = 3.1667
- (3.1667 + 10 ÷ 3.1667) ÷ 2 = 3.1623
- After two rounds you already have √10 ≈ 3.1623, correct to four decimal places.
Radicals made simple
How to Simplify Square Roots
To simplify a square root, pull every perfect-square factor out of the radical.
- Write the number as a product of prime factors.
- Circle each pair of identical factors.
- Move one factor from each pair outside the root and multiply them.
- Multiply the unpaired factors and leave them inside.
Here are some common examples:
- √18 = 3√2, because 18 = 9 × 2
- √48 = 4√3, because 48 = 16 × 3
- √50 = 5√2, because 50 = 25 × 2
- √72 = 6√2, because 72 = 36 × 2
- √200 = 10√2, because 200 = 100 × 2
Beyond square roots
Cube Roots and nth Roots
A cube root undoes cubing, so ∛64 = 4 because 4 × 4 × 4 = 64. The same simplification method works, except you group factors in threes instead of pairs. For example, 108 = 2² × 3³, so ∛108 = 3∛4.
In general, the nth root groups factors into sets of n. For instance, the 4th root of 48 is 2⁴√3, because 48 = 2⁴ × 3. Every root can also be written as a power: the nth root of x equals x to the power 1/n.
Special cases
Square Roots of Fractions, Decimals and Negative Numbers
Fractions
Take the root of the top and bottom separately: √(9/16) = 3/4. When the bottom is not a perfect square, multiply to clear it. For example, √(2/3) = √6 / 3.
Decimals
Turn the decimal into a fraction first. So √0.25 = √(1/4) = 1/2, which is 0.5.
Negative numbers
No real number squares to a negative, so square roots of negatives use the imaginary unit i. As a result, √(−16) = 4i. Odd roots are different, since a negative number cubed stays negative: ∛(−125) = −5.
Benefits
Who Uses a Square Root Calculator?
Students
Check algebra homework, learn to simplify radicals and practice estimating roots.
Teachers and tutors
Show factor grouping on screen and share links that open the exact same problem.
Builders and engineers
Work out diagonals and lengths with the Pythagorean theorem, which relies on square roots.
Data and finance
Standard deviation, volatility and growth rates all involve roots behind the scenes.
Questions
Square Root Calculator FAQs
What is the square root of a number?
The square root of x is the number that gives x when multiplied by itself. For example, the square root of 49 is 7 because 7 × 7 = 49. Every positive number has two square roots, one positive and one negative, and the √ symbol refers to the positive one.
How do you simplify a square root?
Break the number into prime factors, group identical factors in pairs, and move one factor from each pair outside the root. For example, 72 = 2 × 2 × 2 × 3 × 3, so √72 = 2 × 3 × √2 = 6√2.
What is the square root of 2?
The square root of 2 is about 1.41421356. It is irrational, so its decimal never ends or repeats, and it cannot be simplified any further.
How do you find a square root without a calculator?
First find the two perfect squares around your number to make an estimate. Then improve the guess with the Babylonian method: average the guess with the number divided by the guess. Two or three rounds usually give several correct decimal places.
Can you take the square root of a negative number?
Not with real numbers, because any real number squared is zero or positive. With imaginary numbers, √(−16) = 4i, where i is the square root of −1. Odd roots are different: the cube root of −27 is −3.
What is a perfect square?
A perfect square is a whole number whose square root is also a whole number, such as 1, 4, 9, 16, 25 and 36. Numbers like 2, 10 and 72 are not perfect squares, so their square roots are irrational.
What is the difference between a square root and a cube root?
A square root undoes squaring and a cube root undoes cubing. So √64 = 8 because 8² = 64, while ∛64 = 4 because 4³ = 64.
How do you find the square root of a fraction or decimal?
Take the root of the top and bottom separately, so √(4/9) = 2/3. For a decimal, turn it into a fraction first: √0.25 = √(1/4) = 1/2 = 0.5.
Try it now
Simplify Your Next Root in Seconds
Enter any number above to get the exact radical form, the decimal and every step. Then try a cube root or a fraction to see the method in action.
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