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العربية

حاسبة الأسس والجذور

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.

آخر تحديث أمثلة تم التحقق منها: 9

Integers, decimals, fractions (27/8) or expressions
A fraction like 2/3 means the cube root, squared
جرّب
النتيجة
النتيجة: 6.349604207873
الحد الأقصى للمنازل العشرية: 12؛ إلى الأقرب، وعند التعادل بعيدًا عن الصفر
Exact form
4∛4

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

y = t^(2/3)

2468-20-1001020ty16 → 6.3496
طريقة الحساب 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

حول حاسبة الأسس والجذور

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.

أمثلة محلولة

8^(2/3)

Calculate
Power xʸ
Base x
8
Exponent y
2/3
النتيجة
4
Exact form
4

مصدر التحقق: ⁨∛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
النتيجة
3.779763149685
Exact form
3∛2

مصدر التحقق: ⁨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
النتيجة
0.125
Exact form
1/8

مصدر التحقق: ⁨Python Fraction(2) ** -3 = 1/8⁩

Negative base, odd root: (−8)^(1/3)

Calculate
Power xʸ
Base x
-8
Exponent y
1/3
النتيجة
-2
Exact form
−2

مصدر التحقق: ⁨(−2)³ = −8 (real cube root)⁩

الأسئلة

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.

ما مدى دقة «⁨حاسبة الأسس والجذور⁩»؟

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 «⁨8^(2/3)⁩» بالرجوع إلى ⁨∛8 = 2, 2² = 4 (hand calculation; Python 8 ** (2/3) = 3.9999999999999996)⁩.

ما مصدر هذه الطريقة؟

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

حول هذه الحاسبة

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}

المصادر

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

تم التحقق بالرجوع إلى المصادر

تتضمن هذه الحاسبة أمثلة محلولة بإجابات من مصادر مستقلة، وعددها 9. تُشغّل ضمن مجموعة الاختبارات، ويمكنك تشغيلها هنا أيضًا.

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