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Square Root Calculator
Enter a number and a root index to get the decimal value and, for whole numbers, the simplified radical form such as 6√2. Cube roots and higher roots are included, and negatives are handled correctly.
Roots and what they undo
The n-th root of a number x is the value that gives x when raised to the power n. The square root of 49 is 7 because 7 × 7 = 49, and the cube root of 27 is 3 because 3 × 3 × 3 = 27. Every positive number has two square roots, one positive and one negative, but the radical sign √ means the principal (non-negative) one. That is why √49 = 7 while the equation x² = 49 has solutions x = 7 and x = −7.
Roots are exponents in disguise: the n-th root of x equals x1/n. The exponent calculator applies the same idea to any power.
Simplifying a radical
A radical is simplified when no factor inside it is a perfect power of the index. The method is to factor the number into primes and pull out each prime that appears at least n times. For √72 = √(23 × 32), the 32 gives a 3 outside, and one pair of 2s gives a 2 outside, leaving a single 2 under the root: 6√2. Check it by squaring: 36 × 2 = 72.
The same approach handles cube roots: ∛54 = ∛(2 × 33) = 3∛2. Simplified form keeps the answer exact, which matters in algebra and geometry where a rounded decimal would carry error into later steps.
Negative numbers
An odd root of a negative number is a real negative number: the cube root of −27 is −3, because (−3)3 = −27. An even root of a negative number is not real. For the square root case, mathematicians define the imaginary unit i with i² = −1, so √(−9) = 3i. This page returns that imaginary result for square roots. For fourth and higher even roots of negatives the answer is a complex number with both a real and imaginary part, and the page reports an error instead of giving a misleading real value.
Estimating and checking roots
To estimate a square root by hand, find the perfect squares on either side. The square root of 72 lies between 8 (64) and 9 (81), and 72 is closer to 81, so a first guess of 8.5 is sensible; the true value is 8.485281. Perfect squares to remember are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144, and the perfect cubes 1, 8, 27, 64, 125 and 216.
Decimal inputs are accepted, but the radical form is shown only for whole numbers up to 1015. Roots of decimals are still given as decimal values with up to ten digits after the point. If you need a root inside a larger expression, the scientific calculator accepts sqrt and cbrt directly.
Worked example
Find the square root of 72. It is not a perfect square, so it simplifies by pulling out the largest square factor, 36.
Inputs
| Number | 72 |
|---|---|
| Root index (2 = square root, 3 = cube root) | 2 |
Result
Square root of 72.
| Decimal value | 8.4852813742 |
|---|---|
| Simplified radical form | 6√2 |
| Perfect square | No |
| Check: result raised to 2 | 72 |
| Nearest perfect powers | 64 (8^2) and 81 (9^2) |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
What is the square root of a number?
It is the non-negative value that, multiplied by itself, gives the original number. For example, the square root of 81 is 9 because 9 × 9 = 81.
How do I simplify a square root?
Factor the number into primes and move each pair of identical primes outside the root. For 72 = 2 × 2 × 2 × 3 × 3, one pair of 2s and one pair of 3s come out, giving 6√2.
Can you take the square root of a negative number?
Not within the real numbers. In the complex numbers, the square root of a negative number is an imaginary number, so the square root of −16 is 4i.
How do I find a cube root?
Set the root index to 3. The cube root of a number is the value that, multiplied by itself three times, returns the number. Unlike square roots, cube roots of negatives are real.
What is a perfect square?
A perfect square is the square of a whole number, such as 1, 4, 9, 16 or 25. Its square root is itself a whole number and needs no radical sign.
Related calculators
References and sources
- NIST Digital Library of Mathematical Functions, section 4.2: powers and roots. Link
- Stewart, J., Precalculus: Mathematics for Calculus, chapter on radicals and rational exponents.