# Máy tính lũy thừa và căn

> 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.

Phiên bản tương tác: https://www.calcopenly.com/vi/math/exponent-root-calculator
Chủ đề: Máy tính toán học

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.

## Dữ liệu đầu vào

- **Calculate** (lựa chọn: 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**

## Kết quả

- Kết quả — kết quả chính
- Exact form

## Công thức

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

## Ví dụ có lời giải

### 8^(2/3)

- Calculate: Power xʸ
- Base x: 8
- Exponent y: 2/3
- **Kết quả: 4**
- **Exact form: 4**
- Nguồn đối chiếu: ∛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
- **Kết quả: 3.779763149685**
- **Exact form: 3∛2**
- Nguồn đối chiếu: 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
- **Kết quả: 0.125**
- **Exact form: 1/8**
- Nguồn đối chiếu: Python Fraction(2) ** -3 = 1/8

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

- Calculate: Power xʸ
- Base x: -8
- Exponent y: 1/3
- **Kết quả: -2**
- **Exact form: −2**
- Nguồn đối chiếu: (−2)³ = −8 (real cube root)

### Square root of 50

- Calculate: Root ⁿ√x
- Number x: 50
- Root n: 2
- **Kết quả: 7.071067811865**
- **Exact form: 5√2**
- Nguồn đối chiếu: 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
- **Kết quả: 2.25**
- **Exact form: 9/4**
- Nguồn đối chiếu: ∛(27/8) = 3/2, (3/2)² = 9/4 (Python fractions)

## Câu hỏi

### 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.

### “Máy tính lũy thừa và căn” chính xác đến mức nào?

Độ chính xác phụ thuộc vào dữ liệu nhập và giả định của phương pháp. Phép tính thập phân dùng 50 chữ số có nghĩa, nhưng ước lượng, phương pháp số và dữ liệu nguồn có thể kém chính xác hơn; làm tròn khi hiển thị không loại bỏ các giới hạn đó. Ví dụ có lời giải đã đối chiếu với nguồn độc lập: 9. Ví dụ, “8^(2/3)” được kiểm tra bằng ∛8 = 2, 2² = 4 (hand calculation; Python 8 ** (2/3) = 3.9999999999999996).

### Phương pháp này lấy từ đâu?

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

## Nguồn

- [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)