Square Root and Nth Root Calculator
Enter a number and which root you want. The default is the square root. You get the decimal answer, the simplest radical form such as 6√2, whether it is a perfect power, and the nearest perfect powers either side.
Odd roots of negative numbers are real. Even roots of negative numbers are not, so the square root of a negative number is given with the imaginary unit i.
Worked examples
A square root in radical form
72, root 2
6√2, about 8.485281374
72 = 36 × 2, and the square root of 36 is 6.
A cube root
54, root 3
3∛2, about 3.779763149
54 = 27 × 2, and the cube root of 27 is 3.
About Square Root and Nth Root Calculator
The nth root of a number is the value that gives the number when multiplied by itself n times. The square root of 49 is 7, the cube root of 27 is 3 and the 4th root of 81 is 3. This calculator finds any root from the square root upwards, gives the decimal to as many places as you ask for, and tells you straight away whether the number is a perfect power such as a square or a cube.
When the number is not a perfect power, the neatest exact answer is often the simplified radical, in which you take out the largest perfect power. The square root of 72 is 6 times the square root of 2, because 72 is 36 times 2. The page finds the prime factors of the number, groups them, and shows you which come out of the root and which stay inside. Odd roots of negative numbers work, so the cube root of -27 is -3, and an even root of a negative number is given with the imaginary unit i.
How to use Square Root and Nth Root Calculator
- Type the number into the first box. It can be a whole number or a decimal.
- Set the root: 2 for a square root, 3 for a cube root, or any whole number up to 1,000.
- Choose how many decimal places to show.
- Read the decimal value, the simplest radical form, the perfect power check and the prime factor working.
Common uses
- Simplifying surds in algebra homework
- Finding the side of a square from its area
- Finding the edge of a cube from its volume
- Checking whether a number is a perfect square or cube
Frequently asked questions
What is a perfect square or perfect cube?
A perfect square is the result of multiplying a whole number by itself, such as 1, 4, 9, 16 and 25. A perfect cube comes from multiplying a whole number by itself twice, such as 1, 8, 27, 64 and 125. Their roots are whole numbers.
What is a simplified radical?
It is the exact form of a root with the largest perfect power taken outside. The square root of 50 is 5 times the square root of 2, written 5√2, which is neater and more useful in algebra than 7.0710678 and so on.
Can I find the square root of a negative number?
Not as a real number. The page gives the answer with the imaginary unit i, so the square root of -9 is 3i. A cube root of a negative number is real, so the cube root of -8 is -2.
How are the decimal digits worked out?
For whole numbers, the digits come from exact whole-number arithmetic, so every digit shown is correct rather than rounded from a short calculator display.
How do I find a fourth or fifth root?
Change the root box. A root of 4 gives the 4th root, 5 gives the 5th root, and so on. The 5th root of 32 is 2.