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.
업데이트 검증한 예제: 9
The square root of 72 is about 8.48528137, which simplifies exactly to 6√2. It lies between 8 and 9 because 72 lies between 64 and 81.
8.4852813742 3857029281 0132345258
Truncated after 30 decimal places; the next digit is 1.
| Prime | Exponent | Groups of 2 (come out) | Left inside |
|---|---|---|---|
| 2 | 3 | 1 | 1 |
| 3 | 2 | 1 | 0 |
Each prime's exponent is split into complete groups of 2 and a leftover.
No prime appears 2 times in 2, so the radical is fully simplified.
Computed with exact integer arithmetic; the digits panel shows 30 decimal places.
72 lies strictly between two consecutive perfect squares, so its square root is between 8 and 9 and is irrational.
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.
검증 출처: OpenStax Elementary Algebra 2e §9.2: √72 = 6√2; Python decimal (60 digits) Decimal(72).sqrt() = 8.4852813742385702928…
검증 출처: calculator.net root calculator example: √27 = 5.196 to 3 places; OpenStax §9.2: √27 = 3√3
검증 출처: Omni Calculator square root page: √52 = 2√13 ≈ 7.22 (7.2111025509… by Python Decimal(52).sqrt())
검증 출처: 54 = 2 × 3³; Python decimal: Decimal(54) ** (Decimal(1) / 3) = 3.7797631496846193…
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).
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.
√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.
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.
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.
정확도는 입력값과 계산 방법의 가정에 따라 달라집니다. 십진 연산은 유효숫자 50자리를 사용하지만, 추정값·수치해석 방법·원본 데이터의 정밀도는 더 낮을 수 있습니다. 표시값을 반올림해도 이러한 한계는 사라지지 않습니다. 독립적인 출처의 풀이와 대조한 계산 예시: 9. 예를 들어 “√72 (default)”은 OpenStax Elementary Algebra 2e §9.2: √72 = 6√2; Python decimal (60 digits) Decimal(72).sqrt() = 8.4852813742385702928…와 대조해 확인합니다.
OpenStax, Elementary Algebra 2e, §9.2 Simplify square roots (√72 = 6√2, √27 = 3√3); Wolfram MathWorld: Square root; Wolfram MathWorld: Pythagoras's constant (digits of √2).
이 계산기에는 독립적인 출처에서 답을 얻은 계산 예제가 9개 있습니다. 테스트 모음에서 실행되며 여기에서도 실행할 수 있습니다.
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