About Square Root Calculator
This square root calculator is free, needs no sign-up and runs in your browser, so the numbers you type are never uploaded. It gives the square root in two forms at once: the exact simplified radical, such as √72 = 6√2, and a decimal to as many places as you choose, from 0 to 100.
The same page handles cube roots, nth roots with any index from 2 to 100, and a dedicated Simplify radical mode. Decimals are computed with exact integer arithmetic rather than floating point, so every digit shown is correct and the rounding is exact. Inputs can be whole numbers, decimals like 2.25 or fractions like 9/16.
Negative numbers are handled honestly. An odd root keeps the sign (the cube root of -8 is -2). An even root of a negative number is not a real number, so the calculator gives the imaginary result instead: √-49 = 7i.
How to use Square Root Calculator
- Choose Square root, Cube root, nth root or Simplify radical.
- Type the number. A whole number, a decimal or a fraction all work, and negatives are allowed.
- For an nth root, type the root index. Choose how many decimal places you want.
- Read the exact form and the decimal at the top.
- Open the steps to see the prime factorization and the pairs taken out of the root.
Features
- Simplified radical form with the prime factorization shown: √72 = 6√2.
- Decimal to up to 100 places, computed exactly with integers.
- Square roots, cube roots and nth roots up to index 100.
- Imaginary results for negative numbers under even roots, and real roots for odd ones.
- Perfect squares and perfect powers are recognised and shown exactly.
- Decimals and fractions accepted, with the denominator rationalised.
- Step-by-step working in KaTeX and a copy button.
How a radical is simplified
Write the number as a product of primes. Every prime that appears twice (for a square root) leaves the root as one copy; whatever is left stays inside. For an nth root, a prime must appear n times to come out.
In symbols, √(a²b) = a√b. For fractions, the denominator is rationalised: √(n/d) = √(n·d) / d.
√72 = √(2³ × 3²) = √(2² × 3² × 2) = 2 × 3 × √2 = 6√2
√x² = |x|
√(-x) = i√xWorked examples
√72 gives 6√2 ≈ 8.485281
72 = 2³ × 3². The pair of 2s and the pair of 3s come out, leaving one 2: 2 × 3 × √2.
√50 gives 5√2 ≈ 7.07106781
50 = 2 × 5², so one 5 comes out.
√2.25 gives 3/2
2.25 is 9/4, and both 9 and 4 are perfect squares, so the root is exact: 3/2 = 1.5.
∛54 gives 3∛2 ≈ 3.779763
54 = 2 × 3³, and three 3s make a cube, so one 3 comes out of the cube root.
√-49 gives 7i
The root of a negative number under an even root is imaginary: √-49 = 7i.
Perfect squares to remember
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 11 | 121 |
| 2 | 4 | 12 | 144 |
| 3 | 9 | 13 | 169 |
| 4 | 16 | 14 | 196 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 16 | 256 |
| 7 | 49 | 17 | 289 |
| 8 | 64 | 18 | 324 |
| 9 | 81 | 19 | 361 |
| 10 | 100 | 20 | 400 |
Good to know
Limits
Radical simplification needs the number to be factored, which is done for numbers up to 24 digits (after rationalising a fraction). Larger numbers still get an exact decimal, but not a simplified radical.
A negative number under an even root above 2 has no real value. The calculator shows the principal complex root as a decimal and does not give an exact radical for it.
Privacy: how it works
- Everything is calculated by code running in your browser tab. The numbers you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- To estimate a square root by hand, find the two perfect squares around it. √50 is between √49 = 7 and √64 = 8, so it is a little over 7.
- A decimal that neither ends nor repeats means the root is irrational, which is true of the root of any number that is not a perfect power.
- Check a decimal root by squaring it. The result should be very close to the number you started with.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I simplify a square root?
Factor the number into primes, then take out one copy of every prime that appears twice. √72 = √(2² × 3² × 2) = 6√2.
What is the square root of a negative number?
There is no real square root. The result is imaginary: √-n = i√n, where i is the square root of -1. So √-49 = 7i.
Is the decimal exact?
Every digit shown is exact and the last one is correctly rounded, because the digits come from integer arithmetic. Irrational roots still continue forever, so the decimal is a rounded value.
What is the difference between a square root and a cube root?
A square root is a number that multiplied by itself gives the original. A cube root is a number that multiplied by itself three times gives it. Cube roots of negative numbers are real: ∛-8 = -2.
Can I take the nth root for any n?
Yes, from 2 up to 100. Use the nth root mode and type the index.
Why does √(x²) equal |x| and not x?
The square root symbol means the non-negative root. If x is negative, x² is positive and its principal root is -x, which is the absolute value |x|.
Can I enter a fraction?
Yes. Type 9/16 and you get 3/4. If the fraction is not a perfect square, the denominator is rationalised in the exact form.
Popular Square Root Calculator pages
- Cube Root CalculatorEnter a number to get its cube root, exact where possible, with the working.
- Nth Root CalculatorEnter a number and a root index to get the nth root with the working.
- Simplify Radical CalculatorEnter the number under the root and get the radical in simplest form with the steps.
- Square Root of 2The square root of 2 is about 1.41421356, and its digits never end or repeat.
- Square Root of 3The square root of 3 is about 1.7320508, and its digits never end or repeat.
- Square Root of 72The square root of 72 simplifies to 6√2, which is about 8.4853.