How to use the root calculator
- Pick square root, cube root or nth root.
- Enter the radicand, the number under the root sign. Negative numbers work with odd roots such as the cube root.
- For an nth root, enter the root index n, a whole number from 2 to 100.
- Read the decimal answer, the simplified radical form for whole numbers, and the working.
How roots are calculated
The index n says which power the root undoes, and the radicand x is the number you start from. The square root of 72 is the number that gives 72 when multiplied by itself. Any root can also be written as a fractional exponent, so √72 = 72^(1/2) and ∛27 = 27^(1/3).
To simplify a radical, the calculator factors the radicand into primes and pulls out every complete group of n equal factors. Each group comes out of the root as a single factor, and anything left over stays inside.
Worked example
The calculator opens with √72:
- Prime factorization: 72 = 2³ × 3².
- Square roots take out pairs. One pair of 2s and one pair of 3s come out: 2 × 3 = 6. One 2 is left inside.
- So √72 = √36 × √2 = 6√2.
- As a decimal: 6 × √2 = 6 × 1.414213562 = 8.485281374.
- Check: 8.485281374² ≈ 72. It isn't exactly 72 only because the decimal was rounded to ten digits.
A quick sanity check also helps. 72 sits between the perfect squares 64 (8²) and 81 (9²), so its square root has to be between 8 and 9.
Perfect squares and cubes to know
| n | n² | n³ |
|---|---|---|
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 6 | 36 | 216 |
| 7 | 49 | 343 |
| 8 | 64 | 512 |
| 9 | 81 | 729 |
| 10 | 100 | 1,000 |
| 12 | 144 | 1,728 |
Knowing these makes simplifying radicals by hand much faster. Look for the largest perfect square that divides the number. For √200 that is 100, so √200 = 10√2.
Roots of negative numbers
An odd root of a negative number is a negative real number: ∛(-27) = -3, because (-3)³ = -27. An even root of a negative number has no real answer, because squaring any real number gives a positive result (or zero). In complex numbers, √(-16) = 4i, where i is the imaginary unit with i² = -1. The calculator gives that value in its message but sticks to real numbers for the answer.
Frequently asked questions
Does a number have two square roots?
Every positive number has two square roots, one positive and one negative: both 6 and -6 square to 36. The √ symbol means the positive one, called the principal square root, which is what the calculator shows first. The other root is listed alongside it.
What does simplest radical form mean?
A radical is in simplest form when no perfect square (or perfect cube for a cube root) divides the number under the root, and there is no fraction under the sign. √50 becomes 5√2 because 50 = 25 × 2. √30 is already simplest, because 30 = 2 × 3 × 5 has no repeated factor.
How do I estimate a square root without a calculator?
Find the two perfect squares around your number. √50 is a little above √49 = 7. For a closer guess, add the distance from 49 divided by twice 7: 7 + 1/14 ≈ 7.07, close to the true 7.0711. This is one step of Newton's method.
Can the index be any number?
This calculator takes whole-number indexes from 2 to 100, which covers what radicals are used for. A root with any other index is a power with a fractional exponent. Use the exponent calculator for something like x^(1/2.5).
Related calculators
- Exponent Calculator — Raise any number to any power, including negative and fractional exponents, or solve for the exponent or the base.
- Prime Factorization Calculator — Break a number into prime factors with exponents, see the factor tree, check if it is prime and count its divisors.
- Pythagorean Theorem Calculator — Find the hypotenuse or a missing leg of a right triangle with a² + b² = c², shown step by step.
- Scientific Calculator — Free online scientific calculator with trig in degrees or radians, logs, powers, roots, factorials, memory and history.
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