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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.

Обновлено Проверенные примеры: 9

Integers, decimals, fractions (27/8) or expressions
A fraction like 2/3 means the cube root, squared
Попробовать
Результат
Результат: 6.349604207873
Максимум знаков после запятой: 12; До ближайшего, при равенстве — от нуля
Exact form
4∛4

16 to the power 2/3 is 4∛4 ≈ 6.34960420787.

y = t^(2/3)

2468-20-1001020ty16 → 6.3496
Как выполняется расчёт S

  1. Fractional exponent as a root

    1623=162316^{\frac{2}{3}} = \sqrt[3]{16^{2}}

    The denominator 3 is the root; the numerator 2 is the power.

  2. Raise to the power first

    162=25616^{2} = 256

  3. Take out perfect cubes

    2563=43⋅43=443\sqrt[3]{256} = \sqrt[3]{4^{3} \cdot 4} = 4\sqrt[3]{4}

  4. Decimal value

    443≈6.3496042078727978994\sqrt[3]{4} \approx 6.349604207872797899

О калькуляторе: Калькулятор степеней и корней

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.

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)

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.

Об этом калькуляторе

xm/k=(xk)m=xmkakrk=ark\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}

Источники

  1. Khan Academy — Rational exponents and radicals
  2. Wolfram MathWorld — Radical

Проверено по источникам

В калькулятор включены решённые примеры с ответами из независимых источников. Их количество: 9. Они входят в набор тестов, и вы также можете запустить их здесь.

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