Root Calculator
Find the square root, cube root or any n-th root of a number. Get an exact answer for perfect powers, the simplest radical form such as √72 = 6√2, and the decimal value, with the prime-factor working.
How to Use
- Type the number. Whole numbers, decimals, fractions such as 8/9 and negatives (for odd roots) all work.
- Choose the root: 2 for a square root, 3 for a cube root, or any whole number up to 100.
- Read the decimal value, the simplest radical form and whether the root is exact.
- Show Work gives the prime factors and which groups come out of the root.
- The graph shows the root curve with your number on it.
Worked Example
Simplifying √72. 72 = 2³ × 3². Each pair comes out of the square root as one: a pair of 2s gives 2 and the pair of 3s gives 3, leaving a single 2 inside. So √72 = 6√2 ≈ 6 × 1.41421 = 8.48528.
A cube root. 54 = 2 × 3³. The three 3s come out as one 3, so ∛54 = 3∛2 ≈ 3.77976.
The common mistake: splitting a root across a sum. √(9 + 16) is not √9 + √16 = 7. Add first: √25 = 5. Roots split across multiplication (√(9 × 16) = 3 × 4 = 12) but never across addition.
Show Work
Formulas
The Square Root of Two
A Babylonian clay tablet, YBC 7289 (about 1800–1600 BC), shows a square with its diagonal and the value of √2 in base 60 as 1;24,51,10, which is 1.41421296, correct to about six decimal places. The method behind it, averaging a guess with the number divided by the guess, is the same Newton’s method computers use today.
The Pythagoreans found that √2 can’t be written as any fraction, the first known irrational number. The √ sign is from the 16th century: Christoph Rudolff used it in 1525, probably from a written letter r for radix, root; Descartes added the bar over the top.
About This Calculator
This calculator finds square, cube and any n-th roots up to the 100th. Perfect powers come out exactly however large they are; other roots are given in simplest radical form, with fractions rationalised, and as a 15-figure decimal. For whole numbers it also shows the perfect powers either side.
Everything runs in your browser; nothing is sent anywhere.
Related calculators: Exponent Calculator, Pythagorean Theorem Calculator, and Prime Factorization Calculator.
Frequently Asked Questions
How do I simplify a square root?
Factor the number and take out every pair. 72 = 2 × 2 × 2 × 3 × 3: one pair of 2s and one pair of 3s come out as 2 × 3 = 6, leaving one 2 inside, so √72 = 6√2. For a cube root take out triples instead: 54 = 2 × 3 × 3 × 3, so ∛54 = 3∛2.
Can I take the square root of a negative number?
Not as a real number, because any real number times itself is positive or zero. The answer is imaginary: √−16 = 4i. Odd roots are different: ∛−27 = −3, because (−3) × (−3) × (−3) = −27.
How do I estimate a square root without a calculator?
Find the perfect squares either side. 50 is between 49 = 7² and 64 = 8², so √50 is just over 7. A better guess: 7 + (50 − 49) ÷ (2 × 7) = 7.071, very close to the true 7.0711. That is one step of Newton’s method, which roughly doubles the correct digits each time.
Why does a positive number have two square roots?
Because (−12)² = 144 as well as 12² = 144. The √ symbol means the positive one, called the principal square root, so √144 = 12. When solving x² = 144 you need both: x = ±12.
What does rationalising the denominator mean?
Rewriting a root so no root is left on the bottom. √(8/9) = √8 ÷ 3 = 2√2/3. For √(1/2) = 1/√2, multiply top and bottom by √2 to get √2/2 ≈ 0.7071. This calculator does it for fractions automatically.
How do I use the Root Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Homework
√72 = 6√2 and ∛54 = 3∛2, with the prime factors.
Geometry
A square of 50 m² has sides of √50 = 7.07 m.
Diagonals
A 1 × 1 square has a diagonal of √2 = 1.414.
Volume
A 1,000 L (1 m³) cube tank is ∛1 = 1 m on each side.
Statistics
A variance of 225 gives a standard deviation of √225 = 15.
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