双重根号

有双重根号的表示式在根号下还有根号，如：${\displaystyle {\sqrt[{5}]{1+{\sqrt {2}}+{\sqrt {3}}}}}$在5次根号下有3个2次根号项。

如果m次根号内的表示式是由一个含根号的多项式自乘m次得来的，都可以化简。

公式

a²-b为平方数时就可以化简${\displaystyle {\sqrt {a\pm {\sqrt {b}}}}}$。

${\displaystyle {\sqrt {a\pm {\sqrt {b}}}}={\frac {{\sqrt {a+{\sqrt {a^{2}-b}}}}\pm {\sqrt {a-{\sqrt {a^{2}-b}}}}}{\sqrt {2}}}}$

例如：${\displaystyle {\sqrt {2+{\sqrt {3}}}}={\frac {{\sqrt {3}}+1}{\sqrt {2}}}={\frac {{\sqrt {6}}+{\sqrt {2}}}{2}}}$

配方法

${\displaystyle {\sqrt {1+2{\sqrt[{5}]{2}}+{\sqrt[{5}]{4}}}}={\sqrt {(1+{\sqrt[{5}]{2}})^{2}}}=1+{\sqrt[{5}]{2}}}$

${\displaystyle {\sqrt[{3}]{5-12{\sqrt[{3}]{3}}+6{\sqrt[{3}]{9}}}}={\sqrt[{3}]{(2)^{3}-3(2)^{2}{\sqrt[{3}]{3}}+3(2){\sqrt[{3}]{9}}-3}}=2-{\sqrt[{3}]{3}}}$

增乘法

对于${\displaystyle {\sqrt[{m}]{{\sqrt {a}}\pm {\sqrt {b}}}}}$，设${\displaystyle x_{1}={\sqrt[{m}]{{\sqrt {a}}+{\sqrt {b}}}},x_{2}={\sqrt[{m}]{{\sqrt {a}}-{\sqrt {b}}}}}$

找x₁+x₂时需要用到${\displaystyle x_{1}^{m}+x_{2}^{m}=\sum _{r=0}^{\lfloor {\frac {m}{2}}\rfloor }{\frac {mC_{m-r}^{r}}{m-r}}(x_{1}+x_{2})^{m-2r}(-x_{1}x_{2})^{r}}$^[1]

${\displaystyle x={\frac {x_{1}+x_{2}\pm {\sqrt {(x_{1}+x_{2})^{2}-4x_{1}x_{2}}}}{2}}}$

${\displaystyle {\sqrt[{3}]{{\sqrt {27}}-{\sqrt {28}}}}}$

${\displaystyle x_{1}x_{2}={\sqrt[{3}]{27-28}}=-1}$

${\displaystyle (x_{1}+x_{2})^{3}+3(x_{1}+x_{2})=6{\sqrt {3}},x_{1}+x_{2}={\sqrt {3}}u,3u^{3}+3u=6,u=1,x_{1}+x_{2}={\sqrt {3}}}$

${\displaystyle {\sqrt[{3}]{{\sqrt {27}}-{\sqrt {28}}}}={\frac {{\sqrt {3}}-{\sqrt {7}}}{2}}}$

对于${\displaystyle {\sqrt[{m}]{k_{1}\pm k_{2}{\sqrt {a}}\pm k_{3}{\sqrt {b}}+k_{4}{\sqrt {ab}}}}}$，设${\displaystyle x_{1}={\sqrt[{m}]{k_{1}+k_{2}{\sqrt {a}}+k_{3}{\sqrt {b}}+k_{4}{\sqrt {ab}}}},x_{2}={\sqrt[{m}]{k_{1}-k_{2}{\sqrt {a}}-k_{3}{\sqrt {b}}+k_{4}{\sqrt {ab}}}}}$

${\displaystyle {\sqrt {15+10{\sqrt {2}}+8{\sqrt {3}}+6{\sqrt {6}}}}}$

${\displaystyle x_{1}x_{2}={\sqrt {49+20{\sqrt {6}}}}=5+2{\sqrt {6}}}$

${\displaystyle x_{1}+x_{2}={\sqrt {40+16{\sqrt {6}}}}=4+2{\sqrt {6}}}$

${\displaystyle {\sqrt {15+10{\sqrt {2}}+8{\sqrt {3}}+6{\sqrt {6}}}}={\frac {4+2{\sqrt {6}}+{\sqrt {20+8{\sqrt {6}}}}}{2}}=2+{\sqrt {2}}+{\sqrt {3}}+{\sqrt {6}}}$

${\displaystyle {\sqrt[{3}]{55+{\frac {81}{2}}{\sqrt {2}}+33{\sqrt {3}}+{\frac {45}{2}}{\sqrt {6}}}}}$

${\displaystyle x_{1}x_{2}=-{\sqrt[{3}]{485+198{\sqrt {6}}}}=-5-2{\sqrt {6}}}$

${\displaystyle (x_{1}+x_{2})^{3}+3(5+2{\sqrt {6}})(x_{1}+x_{2})=5(22+9{\sqrt {6}}),(22-9{\sqrt {6}})(x_{1}+x_{2})^{3}+3(2-{\sqrt {6}})(x_{1}+x_{2})=-10,x_{1}+x_{2}=(2+{\sqrt {6}})u}$

${\displaystyle -4u^{3}-6u=-10,u=1,x_{1}+x_{2}=2+{\sqrt {6}}}$

${\displaystyle {\sqrt[{3}]{55+{\frac {81}{2}}{\sqrt {2}}+33{\sqrt {3}}+{\frac {45}{2}}{\sqrt {6}}}}={\frac {2+{\sqrt {6}}+{\sqrt {30+12{\sqrt {6}}}}}{2}}={\frac {2+3{\sqrt {2}}+2{\sqrt {3}}+{\sqrt {6}}}{2}}}$

参考资料

1. 何万程. 二次根式开方的化简. 数学空间. 2011, (6) [2014-01-07]. （原始内容存档于2014-01-07）. 

参见

• 等幂求和
• 開根號
• 四則運算