Square Root Calculator
Find the square root, cube root or any nth root of a whole number, decimal or fraction. You get the simplified radical form with every step (√72 = 6√2), a check for perfect squares and the decimal to as many places as you like.
Whole numbers, decimals (0.5), fractions (8/9), negatives (−16), powers (2^101) and big numbers up to 300 digits.
How your root is worked out
Your number: split it into prime factors, take out full groups, then work out the decimal.
- 1
Split into primes
72 = 2³ × 3²
- 2
Group the factors
2² × 3² × 2
- 3
Take groups out
√72 = 6√2
- 4
As a decimal
≈ 8.4852813742
Read more: the formula, a worked example, perfect squares and cubes
A root undoes a power. The square root of x is the number that, multiplied by itself, gives x: √72 × √72 = 72. The cube root multiplies by itself three times, and the nth root n times. Most roots never end as decimals, so the neatest exact answer is the simplified radical form, where every whole part has been taken out of the root.
The formula
nth root of x = x^(1/n), the number r with r^n = x
nth root of (a^n × b) = a × nth root of b (take out each full group of n equal prime factors)
nth root of (a ÷ b) = nth root of a ÷ nth root of b, then multiply top and bottom to clear the root from the bottom
odd root of −x = −(odd root of x); square root of −x = i × square root of x
Newton's method: next guess = ((n − 1) × guess + x ÷ guess^(n − 1)) ÷ n; for square roots, (guess + x ÷ guess) ÷ 2
Worked example
Square root of 72 (the default): the prime factors are 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². Pairs: one pair of 2s and one pair of 3s, with one 2 left over. Each pair comes out of the root as a single number, so 2 × 3 = 6 goes outside and 2 stays inside: √72 = 6√2. Check: (6√2)² = 6² × 2 = 36 × 2 = 72. As a decimal, √2 = 1.41421356237…, so 6√2 = 8.48528137423… = 8.4852813742 to 10 decimal places. It is not a perfect square, and it sits between 8 and 9 because 8² = 64 < 72 < 81 = 9².
Newton's method from 8.5: 72 ÷ 8.5 = 8.470588235, and (8.5 + 8.470588235) ÷ 2 = 8.485294118. Next: 72 ÷ 8.485294118 = 8.485268631, and (8.485294118 + 8.485268631) ÷ 2 = 8.485281374. Two rounds already give 8 correct decimal places.
Cube root of 54: 54 = 2 × 3 × 3 × 3 = 2 × 3³. One group of three 3s comes out as 3, leaving 2 inside: ∛54 = 3∛2 ≈ 3 × 1.2599210499 = 3.7797631497. Check: 3³ × 2 = 27 × 2 = 54.
A fraction: √(8/9) = √8 ÷ √9 = 2√2 ÷ 3 = 2√2/3 ≈ 0.9428090416. For √(1/2) = 1 ÷ √2, multiply top and bottom by √2 to get √2/2 ≈ 0.7071067812.
Perfect squares and cubes
| n | n² (square) | n³ (cube) | n | n² | n³ |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 11 | 121 | 1,331 |
| 2 | 4 | 8 | 12 | 144 | 1,728 |
| 3 | 9 | 27 | 13 | 169 | 2,197 |
| 4 | 16 | 64 | 14 | 196 | 2,744 |
| 5 | 25 | 125 | 15 | 225 | 3,375 |
| 6 | 36 | 216 | 16 | 256 | 4,096 |
| 7 | 49 | 343 | 17 | 289 | 4,913 |
| 8 | 64 | 512 | 18 | 324 | 5,832 |
| 9 | 81 | 729 | 19 | 361 | 6,859 |
| 10 | 100 | 1,000 | 20 | 400 | 8,000 |
Common roots to 10 decimal places
| Root | Value | Root | Value |
|---|---|---|---|
| √2 | 1.4142135624 | ∛2 | 1.2599210499 |
| √3 | 1.7320508076 | ∛3 | 1.4422495703 |
| √5 | 2.2360679775 | ∛4 | 1.5874010520 |
| √6 | 2.4494897428 | ∛5 | 1.7099759467 |
| √7 | 2.6457513111 | ∛10 | 2.1544346900 |
| √10 | 3.1622776602 | √8 = 2√2 | 2.8284271247 |
The long-division method
Split the number into pairs of digits from the decimal point (72 → 72 . 00 00). For each pair, bring it down next to what is left and find the largest digit d where (20 × the root so far + d) × d fits. For 72: 8 × 8 = 64 leaves 8; bring down 00 to get 800; (160 + 4) × 4 = 656 fits, leaving 144; so far 8.4. Each round gives one more exact digit. Cube roots work the same way with groups of three digits and (300p² + 30pd + d²) × d.
Negative numbers and complex roots
An odd root of a negative number is real and negative: ∛−54 = −3∛2. An even root of a negative number has no real answer. For square roots the calculator gives the answer with i (√−72 = 6i√2). For 4th, 6th and higher even roots it gives the principal root, which is the root of the size turned by 180° ÷ n: ∜−16 = 2 × (cos 45° + i sin 45°) = 1.414213562 + 1.414213562i. In Advanced you can see all n roots, spaced evenly round a circle.
How it stays exact: the decimal is the whole-number nth root of your number × 10^(n × places), worked out with whole numbers of any size, so every digit shown is right (√2 to 1,000 places included). The last digit is rounded half up, or cut off if you pick that in Advanced. Simplified forms come from the prime factors; numbers too big to split into primes within a second or two are simplified as far as possible, and the page says so.
Common questions
How do you simplify a square root like √72?
Split the number into prime factors 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, and one 2 is left inside, so √72 = 6√2 ≈ 8.4852813742. For a cube root you take out groups of three: 54 = 2 × 3 × 3 × 3, so ∛54 = 3∛2.
What is the square root of a negative number?
No real number times itself gives a negative number, so the answer uses i, the square root of −1. √−16 = √16 × i = 4i, and √−72 = 6i√2. Odd roots of negatives are real: ∛−8 = −2 because (−2) × (−2) × (−2) = −8.
How do I find the square root of a fraction?
Take the root of the top and the bottom separately: √(9/16) = 3/4. If a root is left on the bottom, multiply top and bottom by it to clear it (rationalising): √(1/2) = 1/√2 = √2/2 ≈ 0.7071067812. A decimal works the same way once you write it as a fraction: √0.5 = √(1/2).
How do you work out a square root by hand?
Find the two whole numbers it sits between (√72 is between 8 and 9, since 64 < 72 < 81), then improve a guess with the Babylonian method: average the guess and the number divided by the guess. From 8.5: (8.5 + 72 ÷ 8.5) ÷ 2 = 8.485294, and one more round gives 8.4852814. The long-division method gives one exact digit per pair of digits instead.
Is the square root of a number positive or negative?
Both 6√2 and −6√2 square to 72, but the √ sign means the positive (principal) root. When you solve x² = 72 you write x = ±√72 to get both answers.
What is a perfect square?
A whole number whose square root is a whole number, like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and 144. In prime factors every exponent of a perfect square is even (144 = 2⁴ × 3²), and every exponent of a perfect cube is a multiple of 3 (216 = 2³ × 3³).
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