Root Calculator
The square root, cube root or any nth root of a number, the simplified radical form such as 6√2, and the real cube root of a negative number.
Result
Principal root: 8.485281
Roots that are whole numbers are shown exactly.
- Simplified radical form
- 6√2
- All real roots
- 8.485281374 and −8.485281374
https://calcgallery.com/algebra-numbers/root-calculator/
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How to use the Root Calculator
- Enter the number whose root you want.
- [object Object]
- Read the root, its simplified radical form, and for even roots, both the positive and negative answers.
The formula
The nth root of x is x raised to the power 1/n: the number that, multiplied by itself n times, gives x.
nth root of x = x^(1/n)Simplified radical form
Writing a root in simplest form means taking out every factor that is a perfect nth power. The page factorises the number into primes, moves each complete group of n identical primes outside the root, and leaves the rest inside: 72 = 2³ × 3², so √72 = 2 × 3 × √2 = 6√2.
Exact whole-number roots
Computers work out roots approximately, so the cube root of 64 can come out as 3.9999999999999996. When the approximate answer rounds to a whole number whose power is exactly the original, the page reports that whole number.
With your numbers
- 72^(1/2) = 8.48528137
Quick reference
Roots that are whole numbers are shown exactly.
| Example | Principal root |
|---|---|
| Square root of 72 | 8.485281 |
| Cube root of 64 | 4 |
| Cube root of −8 | -2 |
| Square root of −4 | |
| Square root of 16 | 4 |
| 50th root of 1 | 1 |
Common questions about the Root Calculator
- How do I simplify a square root?
- Pull out the largest perfect square factor. 72 = 36 × 2, and √36 = 6, so √72 = 6√2. The same works for any root: for a cube root, pull out the largest perfect cube.
- Can you take the cube root of a negative number?
- Yes. (−2)³ = −8, so the cube root of −8 is −2. Every odd root of a negative number is real and negative. Many calculators refuse, because they compute roots through logarithms, which only work for positive numbers; this page does not.
- Why does a square root have two answers?
- Because both 4 × 4 and (−4) × (−4) equal 16. The symbol √16 means the principal, positive root, 4, which is what the page shows first; it lists −4 as well, since equations like x² = 16 have both.
- What about the square root of a negative number?
- It is not a real number, since no real number times itself is negative. It is an imaginary number: √−4 = 2i.