# Калькулятор степеней и корней

> Raise a number to any exponent, including fractional and negative ones, or take its square, cube or nth root, with exact simplified radicals such as 5√2.

Интерактивная версия: https://www.calcopenly.com/ru/math/exponent-root-calculator
Тема: Математические калькуляторы

An exponent says how many times to multiply a base by itself: 2⁵ = 32. The rules extend it beyond whole numbers. A negative exponent is a reciprocal (2⁻³ = 1/8), and a fractional exponent m/k is a root, x^(m/k) = (ᵏ√x)^m, so 8^(2/3) = (∛8)² = 4. Roots are simplified by pulling out perfect powers: √50 = √(25 × 2) = 5√2.

Algebra homework, simplifying radicals and formulas with fractional powers are the usual uses. The default, 16^(2/3), is ∛256 = 4∛4 ≈ 6.3496, and the cube root of 54 is 3∛2 because 54 = 27 × 2.

The answer is exact whenever an exact form exists, as a fraction or a simplified radical, and the decimal value is shown to 12 places. Odd roots of negative numbers are real, (−8)^(1/3) = −2; even roots of negative numbers have no real value. 0⁰ is taken as 1.

## Входные данные

- **Calculate** (варианты: Power xʸ, Root ⁿ√x)
- **Base x**: Integers, decimals, fractions (27/8) or expressions
- **Exponent y**: A fraction like 2/3 means the cube root, squared
- **Number x**
- **Root n**

## Результаты

- Результат — основной результат
- Exact form

## Формула

$$
\begin{gathered} x^{m/k} = \left(\sqrt[k]{x}\right)^{m} = \sqrt[k]{x^{m}} \\[10pt] \sqrt[k]{a^{k} r} = a\sqrt[k]{r} \end{gathered}
$$

## Примеры с решением

### 8^(2/3)

- Calculate: Power xʸ
- Base x: 8
- Exponent y: 2/3
- **Результат: 4**
- **Exact form: 4**
- Источник проверки: ∛8 = 2, 2² = 4 (hand calculation; Python 8 ** (2/3) = 3.9999999999999996)

### Cube root of 54

- Calculate: Root ⁿ√x
- Number x: 54
- Root n: 3
- **Результат: 3.779763149685**
- **Exact form: 3∛2**
- Источник проверки: 54 = 27 × 2; Python decimal (60 digits) Decimal(54) ** (Decimal(1)/3) = 3.7797631496846193…

### 2^−3

- Calculate: Power xʸ
- Base x: 2
- Exponent y: -3
- **Результат: 0.125**
- **Exact form: 1/8**
- Источник проверки: Python Fraction(2) ** -3 = 1/8

### Negative base, odd root: (−8)^(1/3)

- Calculate: Power xʸ
- Base x: -8
- Exponent y: 1/3
- **Результат: -2**
- **Exact form: −2**
- Источник проверки: (−2)³ = −8 (real cube root)

### Square root of 50

- Calculate: Root ⁿ√x
- Number x: 50
- Root n: 2
- **Результат: 7.071067811865**
- **Exact form: 5√2**
- Источник проверки: 50 = 25 × 2; Python Decimal(50).sqrt() at 60 digits = 7.07106781186547524400844362104849…

### (27/8)^(2/3)

- Calculate: Power xʸ
- Base x: 27/8
- Exponent y: 2/3
- **Результат: 2.25**
- **Exact form: 9/4**
- Источник проверки: ∛(27/8) = 3/2, (3/2)² = 9/4 (Python fractions)

## Вопросы

### What does a fractional exponent mean?

The denominator is a root and the numerator is a power: x^(m/k) = (ᵏ√x)^m. So 8^(2/3) takes the cube root of 8, which is 2, and squares it to give 4, while 16^(1/2) = √16 = 4. A decimal exponent works the same way once written as a fraction: 2^0.5 = 2^(1/2) = √2 ≈ 1.41421.

### What does a negative exponent mean?

A negative exponent means the reciprocal of the positive power: x⁻ⁿ = 1/xⁿ. So 2⁻³ = 1/2³ = 1/8 = 0.125, and 10⁻² = 0.01. The rule follows from dividing powers, 2³ ÷ 2⁶ = 2³⁻⁶ = 2⁻³. A negative exponent never makes the result negative, and 0 raised to a negative power is undefined because it means dividing by zero.

### How do you simplify a square root?

Split the number into the largest perfect square times what is left, then take the square root of the square. 50 = 25 × 2, so √50 = 5√2 ≈ 7.0711. Higher roots work the same way with perfect cubes, fourth powers and so on: 54 = 27 × 2, so ∛54 = 3∛2. The radical is fully simplified when nothing left under the sign has such a factor, as with √2 or ∛4.

### Why is 0 to the power 0 equal to 1?

By convention x⁰ = 1 for every x, including 0, because that keeps formulas such as the binomial theorem and the power series eˣ = Σ xⁿ/n! correct at x = 0; Knuth's Concrete Mathematics (§5.1) argues for this definition. As a limit, though, 0⁰ is an indeterminate form: xʸ can approach different values as x and y both approach 0.

### Can you take the square root of a negative number?

Not as a real number, because no real number squared gives −4. Odd roots are different: a negative number cubed stays negative, so ∛(−8) = −2 and the fifth root of −32 is −2. Even roots of negative numbers exist only as complex numbers, √(−4) = 2i, which the complex number calculator handles.

### Насколько точен «Калькулятор степеней и корней»?

Точность зависит от введённых данных и допущений метода. Десятичная арифметика использует 50 значащих цифр, но оценки, численные методы и исходные данные могут быть менее точными; округление на экране не устраняет эти ограничения. Решённые примеры, проверенные по независимым источникам: 9. Например, «8^(2/3)» проверяется по источнику ∛8 = 2, 2² = 4 (hand calculation; Python 8 ** (2/3) = 3.9999999999999996).

### Откуда взята методика?

Khan Academy — Rational exponents and radicals; Wolfram MathWorld — Radical.

## Источники

- [Khan Academy — Rational exponents and radicals](https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:rational-exponents-radicals)
- [Wolfram MathWorld — Radical](https://mathworld.wolfram.com/Radical.html)