거듭제곱과 거듭제곱근 계산기

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.

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.

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.

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.

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.

정확도는 입력값과 계산 방법의 가정에 따라 달라집니다. 십진 연산은 유효숫자 50자리를 사용하지만, 추정값·수치해석 방법·원본 데이터의 정밀도는 더 낮을 수 있습니다. 표시값을 반올림해도 이러한 한계는 사라지지 않습니다. 독립적인 출처의 풀이와 대조한 계산 예시: 9. 예를 들어 “8^(2/3)”은 ∛8 = 2, 2² = 4 (hand calculation; Python 8 ** (2/3) = 3.9999999999999996)와 대조해 확인합니다.

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