# حاسبة الجذر التربيعي وتبسيط الجذور

> 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.

النسخة التفاعلية: https://www.calcopenly.com/ar/math/square-root-calculator
الموضوع: حاسبات الرياضيات

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.

## المدخلات

- **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.

## النتائج

- Root — النتيجة الرئيسية
- Simplest radical form
- Perfect power?
- Negative root

## الصيغة

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

## أمثلة محلولة

### √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**
- مصدر التحقق: ⁨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**
- مصدر التحقق: ⁨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**
- مصدر التحقق: ⁨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: لا ينطبق**
- مصدر التحقق: ⁨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**
- مصدر التحقق: ⁨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**
- مصدر التحقق: ⁨12 × 12 = 144⁩

## الأسئلة

### 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.

### ما مدى دقة «⁨حاسبة الجذر التربيعي وتبسيط الجذور⁩»؟

تعتمد الدقة على مدخلاتك وافتراضات الطريقة. يستخدم الحساب العشري 50 رقمًا معنويًا، لكن التقديرات والأساليب العددية وبيانات المصدر قد تكون أقل دقة؛ تقريب القيم المعروضة لا يزيل هذه الحدود. أمثلة محلولة جرى التحقق منها بمصادر مستقلة: 9. مثلًا، يجري التحقق من «⁨√72 (default)⁩» بالرجوع إلى ⁨OpenStax Elementary Algebra 2e §9.2: √72 = 6√2; Python decimal (60 digits) Decimal(72).sqrt() = 8.4852813742385702928…⁩.

### ما مصدر هذه الطريقة؟

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).

## المصادر

- [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)