有双重根号的表示式在根号下还有根号,如: 1 + 2 + 3 5 {\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。
a2-b为平方数时就可以化简 a ± b {\displaystyle {\sqrt {a\pm {\sqrt {b} 。
a ± b = a + a 2 − b ± a − a 2 − b 2 {\displaystyle {\sqrt {a\pm {\sqrt {b}={\frac {\sqrt {a+{\sqrt {a^{2}-b}\pm {\sqrt {a-{\sqrt {a^{2}-b}{\sqrt {2}
例如: 2 + 3 = 3 + 1 2 = 6 + 2 2 {\displaystyle {\sqrt {2+{\sqrt {3}={\frac {\sqrt {3}+1}{\sqrt {2}={\frac {\sqrt {6}+{\sqrt {2}{2}
1 + 2 2 5 + 4 5 = ( 1 + 2 5 ) 2 = 1 + 2 5 {\displaystyle {\sqrt {1+2{\sqrt[{5}]{2}+{\sqrt[{5}]{4}={\sqrt {(1+{\sqrt[{5}]{2})^{2}=1+{\sqrt[{5}]{2}
5 − 12 3 3 + 6 9 3 3 = ( 2 ) 3 − 3 ( 2 ) 2 3 3 + 3 ( 2 ) 9 3 − 3 3 = 2 − 3 3 {\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}
对于 a ± b m {\displaystyle {\sqrt[{m}]{\sqrt {a}\pm {\sqrt {b} ,设 x 1 = a + b m , x 2 = a − b m {\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 x 1 m + x 2 m = ∑ r = 0 ⌊ m 2 ⌋ m C m − r r m − r ( x 1 + x 2 ) m − 2 r ( − x 1 x 2 ) r {\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]
x = x 1 + x 2 ± ( x 1 + x 2 ) 2 − 4 x 1 x 2 2 {\displaystyle x={\frac {x_{1}+x_{2}\pm {\sqrt {(x_{1}+x_{2})^{2}-4x_{1}x_{2}{2}
27 − 28 3 {\displaystyle {\sqrt[{3}]{\sqrt {27}-{\sqrt {28}
对于 k 1 ± k 2 a ± k 3 b + k 4 a b m {\displaystyle {\sqrt[{m}]{k_{1}\pm k_{2}{\sqrt {a}\pm k_{3}{\sqrt {b}+k_{4}{\sqrt {ab} ,设 x 1 = k 1 + k 2 a + k 3 b + k 4 a b m , x 2 = k 1 − k 2 a − k 3 b + k 4 a b m {\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}
15 + 10 2 + 8 3 + 6 6 {\displaystyle {\sqrt {15+10{\sqrt {2}+8{\sqrt {3}+6{\sqrt {6}
55 + 81 2 2 + 33 3 + 45 2 6 3 {\displaystyle {\sqrt[{3}]{55+{\frac {81}{2}{\sqrt {2}+33{\sqrt {3}+{\frac {45}{2}{\sqrt {6}