Square Root Calculator
Square roots, cube roots and nth roots with the radical simplified (√72 = 6√2) via prime factorization, plus the number squared and cubed.
Square Root Calculator: with the default inputs, root is 8.48528137.
The number under the root sign.
2 = square root, 3 = cube root, and so on.
- Simplest radical form
- 6√2
- Number squared
- 5,184
- Number cubed
- 373,248
Assumptions
- Radical simplification is shown for whole-number radicands up to 10¹².
- Even-degree roots of negative numbers return an error (no real value); odd-degree roots are real.
| k | k² | |
|---|---|---|
| 4 | 16 | |
| 5 | 25 | |
| 6 | 36 | |
| 7 | 49 | |
| 8 | 64 | ◀ 72 sits between this row and the next |
| 9 | 81 | |
| 10 | 100 | |
| 11 | 121 | |
| 12 | 144 | |
| 13 | 169 |
√72 = 8.485281, which is why it lands between 8 and 9. Squeezing a number between two perfect powers is how you check a root without a calculator.
How this is worked out
The formula
ⁿ√x = the number r with rⁿ = x √(a² · b) = a√b (pull perfect squares out of the radical) √(a·b) = √a · √b, √(a/b) = √a / √b Newton's method: r_next = ((n − 1)·r + x / rⁿ⁻¹) ÷ n
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Number (radicand)
- The number under the root sign.defaults to 72
- Root degree
- 2 = square root, 3 = cube root, and so on.from 1 to 100 · whole numbers only · defaults to 2
What you get back
- Rootmain answer
- Simplest radical form
- Number squared
- Number cubed
What this assumes
- Radical simplification is shown for whole-number radicands up to 10¹².
- Even-degree roots of negative numbers return an error (no real value); odd-degree roots are real.
About this calculator
The square root of a number is whatever you square to get it back: √49 = 7 because 7² = 49. A cube root undoes cubing, and an n-th root undoes the n-th power. Most roots aren't whole numbers — √2 = 1.41421356… never terminates — so a good answer often has two parts: the exact radical form and a decimal approximation. This calculator gives both.
How to use it
Type the number and, if you want something other than a square root, the degree (3 for cube root). For whole numbers the calculator factors the radicand into primes and pulls out every complete pair (or triple, for cube roots), which is exactly how "simplest radical form" is produced by hand.
Why √72 = 6√2
72 = 2³ × 3² = (2 × 3)² × 2. A square root undoes squaring, so the (2 × 3)² comes out as 6 and the lone 2 stays inside: 6√2 ≈ 8.485. The same idea works for any degree: ∛54 = ∛(3³ × 2) = 3∛2.
Negative numbers
Squaring anything real gives a positive result, so √−4 has no real answer; the calculator says so and mentions the imaginary answer 2i. Odd roots are different: (−2)³ = −8, so ∛−8 = −2 is a perfectly real number and the calculator returns it.
Which root?
Every positive number has two square roots (7 and −7 both square to 49). The radical sign √ means the principal (non-negative) one; if you need both, as when solving x² = 49, write ±√49.
Common mistakes
- √(a + b) is not √a + √b. √(9 + 16) = 5, not 7.
- Forgetting to check whether a "simplified" radical still contains a perfect square (√8 = 2√2, not "√8 is simplest").
- Rounding a root and then squaring the rounded value to check — expect a small discrepancy in the last digits, not an exact match.
Frequently asked questions
▸How do I simplify a square root?
Factor the number into primes, pull each pair out as a single factor, and leave the rest inside. √72 = √(2² × 3² × 2) = 2 × 3 × √2 = 6√2.
▸Can you take the square root of a negative number?
Not in the real numbers, because no real number squares to a negative. In the complex numbers √−4 = 2i. Odd roots of negatives are fine: ∛−8 = −2.
▸How do I find a square root without a calculator?
Guess, then improve: average your guess with the number divided by the guess and repeat. For √10, start at 3: (3 + 10/3)/2 = 3.1667, then (3.1667 + 10/3.1667)/2 = 3.1623. That's Newton's method and it converges very fast.
▸Is √16 equal to 4 or ±4?
The radical symbol means the principal (positive) root, so √16 = 4. The equation x² = 16 has two solutions, ±4, which is why you write ± in front of the root when solving equations.
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