√ 根式化简计算器（平方根、立方根、n 次根、系数）Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient)

使用说明与常见问题How-to & FAQ

「根式化简计算器（平方根、立方根、n 次根、系数）」是什么？What is the Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient) tool?

把 k·ⁿ√x 化成最简根式 a·ⁿ√b（b 不含 n 次方因数），支持平方根、立方根和任意 n 次根，可带系数，负数的奇次根会保留负号。同时给出质因数分解、提出的因数和小数值。本地处理。Simplifies k·ⁿ√x to simplest radical form a·ⁿ√b (b has no nth-power factor) for square, cube or any nth roots, with an optional coefficient; odd roots of negative numbers keep the minus sign. Also shows the prime factorisation, the factor taken out and the decimal value. Processed locally.

如何使用「根式化简计算器（平方根、立方根、n 次根、系数）」？How do I use the Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient) tool?

在「根式化简计算器（平方根、立方根、n 次根、系数）」中，填写「n（非负整数）」、「根指数」、「系数（可选）」等参数，结果会随输入实时计算并显示在下方。In the Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient): Fill in "n (non-neg int)", "Root index", "Coefficient (optional)"; results are recalculated instantly as you type.

根式怎么化简？How do I simplify a radical?

找出被开方数中最大的完全平方（或 n 次方）因数并移到根号外。例如 √50 = √(25 × 2) = 5√2，∛54 = ∛(27 × 2) = 3∛2。Find the largest perfect square (or nth power) factor of the radicand and move its root outside. √50 = √(25 × 2) = 5√2 and ∛54 = ∛(27 × 2) = 3∛2.

带系数怎么算？What about a coefficient?

提出的因数要乘上原来的系数。例如 3√72 = 3 × √(36 × 2) = 3 × 6√2 = 18√2。Multiply the factor taken out by the existing coefficient: 3√72 = 3 × √(36 × 2) = 3 × 6√2 = 18√2.

负数可以开方吗？Can I take roots of negative numbers?

奇次根可以，结果为负，例如 ∛(−16) = −2∛2。偶次根（如平方根）在实数范围内没有定义。Odd roots can: ∛(−16) = −2∛2. Even roots such as square roots of negatives are not real numbers.

能举个计算实例吗？Can you show a worked example?

例如在「根式化简计算器（平方根、立方根、n 次根、系数）」中填 n（非负整数） = 50，结果区显示：化简：5√2For example, entering n (non-neg int) = 50 in the Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient) gives: Simplified: 5√2

哪些输入会被检查？Which inputs are validated?

「根式化简计算器（平方根、立方根、n 次根、系数）」会检查这些输入——「n（非负整数）」：必须填写，须为整数，绝对值不超过 10¹²；平方根、四次根等偶次根时不能为负；「系数（可选）」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient) checks its inputs — "n (non-neg int)": required, must be a whole number of at most 10¹² in absolute value, not negative for even roots (square, fourth, sixth); "Coefficient (optional)": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「根式化简计算器（平方根、立方根、n 次根、系数）」的设置：「n（非负整数）」默认 50；「根指数」可选 平方根 √、立方根 ∛、四次根 ∜、五次根、六次根（默认 平方根 √）；「系数（可选）」可留空。每次生成结果都按当前设置处理。Settings in the Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient) — "n (non-neg int)" default 50; "Root index": Square root √, Cube root ∛, Fourth root ∜, Fifth root, Sixth root (default Square root √); "Coefficient (optional)" optional. The result always uses the current settings.

计算精度如何？How precise are the results?

「根式化简计算器（平方根、立方根、n 次根、系数）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「根式化简计算器（平方根、立方根、n 次根、系数）」免费吗？需要注册吗？Is the Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient) free? Do I need an account?

「根式化简计算器（平方根、立方根、n 次根、系数）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Simplify Radicals Calculator (Square, Cube and nth Roots, With a Coefficient) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.