√72 (default)
- Number
- 72
- Root
- 2
- Decimal places
- 30
- Root
- 8.48528137424
- Simplest radical form
- 6√2
- Perfect power?
- No: 8² = 64 < 72 < 81 = 9²
- Negative root
- -8.48528137424
Checked against: OpenStax Elementary Algebra 2e §9.2: √72 = 6√2
Square root calculator and radical simplifier
Square root calculator: the root to up to 1,000 decimal places, simplest radical form (√72 = 6√2), a perfect-square check and cube or nth roots.
By AdminDrafted with AI writing toolsUpdated Checked examples: 9
Root
Root: 8.48528137424
- Simplest radical form
- 6√2
- Perfect power?
- No: 8² = 64 < 72 < 81 = 9²
- Negative root
- −8.48528137424
The square root of 72 is about 8.48528137. An exact radical form is 6√2. It lies between 8 and 9 because 72 lies between 64 and 81.
√72 to 30 decimal places
8.4852813742 3857029281 0132345258
Truncated after 30 decimal places; the next digit is 1.
y = √x between consecutive integer powers with exponent 2
Prime factors of 72
| Prime | Exponent | Groups of 2 (come out) | Left inside |
|---|---|---|---|
| 2 | 3 | 1 | 1 |
| 3 | 2 | 1 | 0 |
How it's calculated S
Prime factorization
Each prime's exponent is split into complete groups of 2 and a leftover.
Take out groups of 2
No prime appears 2 times in 2, so the radical is fully simplified.
Decimal value
Computed with exact integer arithmetic; the digits panel shows 30 decimal places.
Is it a perfect square?
72 lies strictly between consecutive integer powers with exponent 2, so its square root is between 8 and 9 and is irrational.
About square root calculator and radical simplifier
The square root of x is the non-negative number that gives x when multiplied by itself; the nth root does the same with n equal factors. The calculator takes the root with exact whole-number arithmetic, so every digit shown is correct, and simplifies the radical by prime factorization: write the number as a product of prime powers, move each complete group of n equal primes outside the root sign, and leave the rest inside.
For 72 = 2³ × 3², one pair of 2s and one pair of 3s come out as 2 × 3 = 6, and one 2 stays inside, so √72 = 6√2 ≈ 8.485281. Since 8² = 64 and 9² = 81, 72 is not a perfect square. Fractions and decimals work too: √0.72 = √(18/25) = 3√2/5.
Digits are truncated, not rounded, and up to 1,000 decimal places are available. An even root of a negative number is not a real number; odd roots of negative numbers are, as ∛(−54) = −3∛2.
How to find a square root
The square root of x is the number that, multiplied by itself, gives x: √144 = 12 because 12 × 12 = 144. A number that is not a perfect square has an irrational root, which simplifies by taking out square factors: √72 = √(36 × 2) = 6√2 ≈ 8.48528137424.
Square roots of common numbers
The decimal column gives the principal square root to twelve significant digits. A decimal input such as 0.25 counts as the exact fraction 1/4.
| Number | Simplest radical form | Principal square root |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | √2 | 1.41421356237 |
| 4 | 2 | 2 |
| 9 | 3 | 3 |
| 16 | 4 | 4 |
| 25 | 5 | 5 |
| 36 | 6 | 6 |
| 49 | 7 | 7 |
| 64 | 8 | 8 |
| 72 | 6√2 | 8.48528137424 |
| 100 | 10 | 10 |
| 144 | 12 | 12 |
| 0.25 | 1/2 | 0.5 |
| 18/25 | 3√2/5 | 0.848528137424 |
| 9/4 | 3/2 | 1.5 |
Larger perfect squares that come up often: √289 = 17, √576 = 24, √676 = 26, √729 = 27, √1024 = 32, √1296 = 36, √1681 = 41, √3844 = 62, √14641 = 121 and √44100 = 210.
Common irrational roots, to twelve significant digits: √3 ≈ 1.73205080757, √5 ≈ 2.2360679775 and √10 ≈ 3.16227766017. Their decimals never end or repeat.
How to simplify a square root
Split the number into the largest perfect square times what is left, then take the root of the square: 90 = 9 × 10, so √90 = 3√10, and 300 = 100 × 3, so √300 = 10√3. The root is fully simplified when nothing left inside has a square factor.
Prime factors find the square factor reliably. Pair up equal primes and move one of each pair outside: 90 = 2 × 3² × 5, so the pair of 3s comes out and √90 = 3√10.
For a fraction, reduce it first, then take the root of top and bottom. 243/867 reduces to 81/289, so √(243/867) = 9/17. Squaring checks a radical answer: (3√2/5)² = 9 × 2 / 25 = 18/25.
When the denominator is not a perfect square, multiply top and bottom to clear the root from it: √(4/7) = 2/√7 = 2√7/7 ≈ 0.755929.
Square root rules and common mistakes
- Products: √(ab) = √a × √b, so √(4 × 9) = 2 × 3 = 6.
- Quotients: √(a/b) = √a ÷ √b, so √(9/4) = 3/2.
- Sums do not split: √(9 + 16) = √25 = 5, not √9 + √16 = 7.
- Squares of negatives: √((−5)²) = √25 = 5, not −5, because √ always returns the nonnegative root.
Square roots turn areas back into lengths. A square room with an area of 50 m² has sides of √50 ≈ 7.07 m, and the diagonal of a 3 m by 4 m rectangle is √(3² + 4²) = 5 m.
How to estimate a square root by hand
Find the perfect squares on either side. Since 81 < 90 < 100, √90 lies between 9 and 10, a little below the midpoint.
The Babylonian method sharpens a guess g to (g + x ÷ g) ÷ 2. From 9.5, one step gives (9.5 + 90 ÷ 9.5) ÷ 2 ≈ 9.48684, against the true value 9.48683298. Each step roughly doubles the number of correct digits.
For √10, starting at 3 gives 3.16667 and then 3.16228, which is √10 correct to five decimal places.
Why a square root has two answers
Every positive number has two square roots, one positive and one negative: 7² = 49 and (−7)² = 49. The symbol √ names the principal, nonnegative one, so √49 = 7, while the equation x² = 49 has the two solutions x = ±7.
A negative number has no real square root, because every real number squared is zero or positive. √(−16) is the complex number 4i, which the complex number calculator handles.
Cube roots and other roots
The root index can be any whole number from 2 to 100: 3 for a cube root, 4 for a fourth root. An odd root of a negative number is real, so ∛(−8) = −2, because (−2)³ = −8. An even root of a negative number is not real.
For roots written as fractional exponents, such as 16^(2/3), use the exponent and root calculator.
Decimal digits are truncated, not rounded
The long decimal expansion runs to as many as 1,000 places and cuts off after the last requested place. Cutting and rounding differ when the next digit is 5 or more: √3 = 1.7320508…, so four places give 1.7320 when cut but 1.7321 when rounded.
More digits never make an irrational root exact. Keep the radical form, such as 6√2, for further algebra, and round only the final decimal.
To list every divisor of a whole number, including its square factors, use the factor calculator.
Worked examples
√27 (calculator.net and OpenStax)
- Number
- 27
- Root
- 2
- Decimal places
- 30
- Root
- 5.196
- Simplest radical form
- 3√3
Checked against: calculator.net root calculator example: √27 = 5.196 to 3 places; OpenStax §9.2: √27 = 3√3
√52 (Omni Calculator example)
- Number
- 52
- Root
- 2
- Decimal places
- 30
- Root
- 7.2111
- Simplest radical form
- 2√13
Checked against: Omni Calculator square root page: √52 = 2√13 ≈ 7.22
Cube root of 54
- Number
- 54
- Root
- 3
- Decimal places
- 30
- Root
- 3.77976314968
- Simplest radical form
- 3∛2
- Perfect power?
- No: 3³ = 27 < 54 < 64 = 4³
- Negative root
- not applicable
Checked against: 54 = 2 × 3³
Questions
How do you simplify a square root?
Factor the number into primes, pair up equal primes, and move one of each pair outside the root. 72 = 2 × 2 × 2 × 3 × 3 has a pair of 2s and a pair of 3s, so 2 × 3 = 6 comes out and one 2 stays in: √72 = 6√2. Equivalently, split off the largest perfect square factor: 72 = 36 × 2, and √36 = 6. A root is simplified when nothing under the sign has a square factor (OpenStax Elementary Algebra 2e, §9.2).
How do you know if a number is a perfect square?
A whole number is a perfect square when every exponent in its prime factorization is even, or equivalently when its square root is a whole number. 144 = 2⁴ × 3² is 12², while 72 = 2³ × 3² is not, because the exponent of 2 is odd. Perfect squares also end only in 0, 1, 4, 5, 6 or 9, so a number ending in 2, 3, 7 or 8, such as 72, can be ruled out at a glance.
What is the square root of 2 to 50 decimal places?
√2 = 1.41421356237309504880168872420969807856967187537694, truncated after 50 decimal places. The digits never end or repeat, because √2 is irrational: it is not a ratio of two whole numbers. The ISO 216 paper sizes use it, so an A4 sheet's sides are in the ratio 297/210 = 1.414.
How do you find the cube root or nth root of a number?
The nth root of x is the number that, used n times as a factor, gives x. To simplify it, take out every complete group of n equal primes: 54 = 2 × 3³, so ∛54 = 3∛2 ≈ 3.779763. For an nth root with a decimal answer, raise x to the power 1/n; calculator.net's example gives the 8th root of 15 as 1.403 to three places.
Why does a positive number have two square roots?
Because a negative times a negative is positive: 8.485…² and (−8.485…)² both equal 72. The radical sign √ means the principal, non-negative root, so √72 = 6√2 and the other root is −6√2; together they are written ±6√2. Solving x² = 72 therefore gives two answers. Odd roots have only one real value, and ∛(−8) = −2.
About this calculator
Sources
Checked against references
This calculator includes 9 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.