# Calculadora de raiz quadrada e simplificador de radicais

> 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.

Versão interativa: https://www.calcopenly.com/pt/math/square-root-calculator
Tema: Calculadoras de matemática

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.

## Dados

- **Number**: A whole number, decimal or fraction, e.g. 72, 0.72 or 18/25.
- **Root**: 2 for the square root, 3 for the cube root, any whole number up to 100 for other roots.
- **Decimal places**: How many digits to show after the decimal point (up to 1,000). Digits are truncated, not rounded.

## Resultados

- Root — resultado principal
- Simplest radical form
- Perfect power?
- Negative root

## Fórmula

$$
\sqrt[n]{a^{n} b} = a\sqrt[n]{b} \qquad \sqrt[n]{\frac{p}{q}} = \frac{\sqrt[n]{p\,q^{\,n-1}}}{q}
$$

## Exemplos resolvidos

### √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**
- Fonte de verificação: OpenStax Elementary Algebra 2e §9.2: √72 = 6√2; Python decimal (60 digits) Decimal(72).sqrt() = 8.4852813742385702928…

### √27 (calculator.net and OpenStax)

- Number: 27
- Root: 2
- Decimal places: 30
- **Root: 5.196**
- **Simplest radical form: 3√3**
- Fonte de verificação: 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**
- Fonte de verificação: Omni Calculator square root page: √52 = 2√13 ≈ 7.22 (7.2111025509… by Python Decimal(52).sqrt())

### 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: não se aplica**
- Fonte de verificação: 54 = 2 × 3³; Python decimal: Decimal(54) ** (Decimal(1) / 3) = 3.7797631496846193…

### 8th root of 15 (calculator.net example)

- Number: 15
- Root: 8
- Decimal places: 30
- **Root: 1.403**
- **Simplest radical form: ⁸√15**
- Fonte de verificação: calculator.net root calculator example: the 8th root of 15 is 1.403 to 3 decimal places

### √144 is a perfect square (edge case)

- Number: 144
- Root: 2
- Decimal places: 30
- **Root: 12**
- **Simplest radical form: 12**
- **Perfect power?: Yes: 144 = 12²**
- Fonte de verificação: 12 × 12 = 144

## Perguntas

### 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.

### Qual é a precisão de “Calculadora de raiz quadrada e simplificador de radicais”?

A precisão depende dos dados inseridos e das hipóteses do método. O cálculo decimal usa 50 algarismos significativos, mas estimativas, métodos numéricos e dados de origem podem ter menor precisão; o arredondamento exibido não elimina essas limitações. Exemplos resolvidos verificados com fontes independentes: 9. Por exemplo, “√72 (default)” é verificado com OpenStax Elementary Algebra 2e §9.2: √72 = 6√2; Python decimal (60 digits) Decimal(72).sqrt() = 8.4852813742385702928….

### De onde vem o método?

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).

## Fontes

- [OpenStax, Elementary Algebra 2e, §9.2 Simplify square roots (√72 = 6√2, √27 = 3√3)](https://openstax.org/books/elementary-algebra-2e/pages/9-2-simplify-square-roots)
- [Wolfram MathWorld: Square root](https://mathworld.wolfram.com/SquareRoot.html)
- [Wolfram MathWorld: Pythagoras's constant (digits of √2)](https://mathworld.wolfram.com/PythagorassConstant.html)