⁵ 五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots)
求 x 的五次方根：负数也有实数五次方根（如 ⁵√−32 = −2）；判断是否为完全五次方，并列出 5 个复数根 r^(1/5)·(cos θ + i sin θ)。本地处理。Find the fifth root of x: negative numbers have a real fifth root too (⁵√−32 = −2). Checks for a perfect fifth power and lists all 5 complex roots r^(1/5)·(cos θ + i sin θ). Processed locally.
使用说明与常见问题How-to & FAQ
「五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）」是什么？What is the Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots) tool?
求 x 的五次方根：负数也有实数五次方根（如 ⁵√−32 = −2）；判断是否为完全五次方，并列出 5 个复数根 r^(1/5)·(cos θ + i sin θ)。本地处理。Find the fifth root of x: negative numbers have a real fifth root too (⁵√−32 = −2). Checks for a perfect fifth power and lists all 5 complex roots r^(1/5)·(cos θ + i sin θ). Processed locally.
如何使用「五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）」？How do I use the Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots) tool?
在「五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）」中，填写「x」等参数，结果会随输入实时计算并显示在下方。In the Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots): Fill in "x"; results are recalculated instantly as you type.
负数能开五次方吗？Can I take the fifth root of a negative number?
能。五次是奇数次，(−2)⁵ = −32，所以 ⁵√−32 = −2。在复数范围内每个非零数都有 5 个五次方根，均匀分布在复平面的圆上。Yes. Five is odd, and (−2)⁵ = −32, so ⁵√−32 = −2. In the complex numbers every non-zero number has 5 fifth roots, evenly spaced on a circle in the complex plane.
能举个计算实例吗？Can you show a worked example?
例如在「五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）」中填 x = -32，结果区显示：实数五次方根：−2；完全五次方？：是：−2⁵ = −32；验算：−2⁵ = −32；全部 5 个复数根：z1 = 1.61803399 + 1.1755705i；z2 = −0.61803399 + 1.90211303i；z3 = −2；z4 = −0.61803399 − 1.90211303i；z5 = 1.61803399 − 1.1755705iFor example, entering x = -32 in the Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots) gives: Real fifth root: −2; Perfect fifth power?: yes: −2⁵ = −32; Check: −2⁵ = −32; All 5 complex roots: z1 = 1.61803399 + 1.1755705i; z2 = −0.61803399 + 1.90211303i; z3 = −2; z4 = −0.61803399 − 1.90211303i; z5 = 1.61803399 − 1.1755705i
哪些输入会被检查？Which inputs are validated?
「五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）」会检查这些输入——「x」：必须填写，须为数字；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 Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots) checks its inputs — "x": required, must be a number; 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?
「五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）」的设置：「x」默认 -32。每次生成结果都按当前设置处理。Settings in the Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots) — "x" default -32. The result always uses the current settings.
计算精度如何？How precise are the results?
「五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）」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 Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）」免费吗？需要注册吗？Is the Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots) free? Do I need an account?
「五次方根计算器（实根、是否为完全五次方、全部 5 个复数根）」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 Fifth Root Calculator (Real Root, Perfect Fifth Power Check, All 5 Complex Roots) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.