FriCAS can denest many nested roots. This is implemented in an extra package so
befor use we need to expose it:
fricas
(1) -> )expose RootSimplification
RootSimplification is now explicitly exposed in frame initial
Now we can try
fricas
r := sqrt(102*sqrt(7) + 272)
\label{eq1}\sqrt{{{102}\ {\sqrt{7}}}+{272}}
(1) fricas
rsimp(r)
\label{eq2}{\left({\sqrt{7}}+ 3 \right)}\ {\sqrt{17}}
(2) Type: Union(Expression(Integer),...)
fricas
r := sqrt(66000162*1000003^(1/2) + 11000159000324)
\label{eq3}\sqrt{{{66000162}\ {\sqrt{1000003}}}+{11000159000324}}
(3) fricas
rsimp(r)
\label{eq4}\frac{{3 \ {\sqrt{\frac{99000243}{1000003}}}\ {\sqrt{1000003}}}+{{1000003}\ {\sqrt{\frac{99000243}{1000003}}}}}{3}
(4) Type: Union(Expression(Integer),...)
fricas
r := sqrt(((-4030008370)*700001^(1/2)+12090025110)*1000003^(1/2)+((-78000162)*700001^(1/2)+312335685800846295))
\label{eq5}\sqrt{{{\left(-{{4030008370}\ {\sqrt{700001}}}+{12090025110}\right)}\ {\sqrt{1 000003}}}-{{78000162}\ {\sqrt{700001}}}+{312335685800846295}}
(5) fricas
rsimp(r)
\label{eq6}\frac{{\left({{155}\ {\sqrt{1000003}}}-{\sqrt{700001}}+ 3 \right)}\ {\sqrt{3 12325648675}}}{155}
(6) Type: Union(Expression(Integer),...)
We can denest roots of higher degree:
fricas
r := (((-6670600520850)*7^(1/2)+73781123539185)*13^(1/2)+((-77245760121201)*7^(1/2)+89932095680661))^(1/7)
\label{eq7}\root{7}\of{{{\left(-{{6670600520850}\ {\sqrt{7}}}+{737811235 39185}\right)}\ {\sqrt{13}}}-{{77245760121201}\ {\sqrt{7}}}+{8 9932095680661}}
(7) fricas
rsimp(r)
\label{eq8}\frac{{\left({5 \ {\sqrt{13}}}-{\sqrt{7}}+ 1 \right)}\ {\root{7}\of{2 1699140625}}}{5}
(8) Type: Union(Expression(Integer),...)
Note: Current code supports root of degree 2, 3 or 4 outside. With square root outside inside can contain
roots of degree up to 12. Roots of degree 3 outside more limited concerning what can appear inside.
And we can have roots of higher degree inside:
fricas
r := (((-827820)*(17^(1/3))^2+(-58105080)*17^(1/3)+(-134584260))*19^(1/2)+(17045208*(17^(1/3))^2+79043889*17^(1/3)+1426984144))^(1/4)
\label{eq9}\root{4}\of{{{\left(-{{827820}\ {{\root{3}\of{17}}^{2}}}-{{58 105080}\ {\root{3}\of{17}}}-{134584260}\right)}\ {\sqrt{19}}}+{{17045208}\ {{\root{3}\of{17}}^{2}}}+{{79043889}\ {\root{3}\of{1 7}}}+{1426984144}}
(9) fricas
rsimp(r)
\label{eq10}\frac{{\left({{15}\ {\sqrt{19}}}-{3 \ {\root{3}\of{17}}}- 7 \right)}\ {\root{4}\of{3 695625}}}{15}
(10) Type: Union(Expression(Integer),...)
Denesting is not limited to numbers, it works for algebraic functions and more general expressions:
fricas
)set output tex off
fricas
)set output algebra on
r := ((x^6+3*x^5+73*x^4+429*x^3+63*x^2+213*x-1287)*(x^3-3)^(1/2)+3*x^7+37*x^6+39*x^5+451*x^4+1545*x^3+315*x^2+351*x-4860)^(1/3)
(11)
ROOT
+------+
6 5 4 3 2 | 3 7
(x + 3 x + 73 x + 429 x + 63 x + 213 x - 1287)\|x - 3 + 3 x
+
6 5 4 3 2
37 x + 39 x + 451 x + 1545 x + 315 x + 351 x - 4860
,
3
Type: Expression(Integer)
fricas
rsimp(r)
+------+ +----------+
| 3 3| 3
(12) (\|x - 3 + x + 12)\|x + x - 3
Type: Union(Expression(Integer),...)
fricas
)set output algebra off
fricas
)set output tex on
r := ((2*exp(x)^2-12*exp(x)-14)*(exp(x)+3)^(1/2)+exp(x)^3-12*exp(x)^2+39*exp(x)+52)^(1/2)
\label{eq11}\sqrt{{{\left({2 \ {{{e}^{x}}^{2}}}-{{12}\ {{e}^{x}}}-{14}\right)}\ {\sqrt{{{e}^{x}}+ 3}}}+{{{e}^{x}}^{3}}-{{12}\ {{{e}^{x}}^{2}}}+{{39}\ {{e}^{x}}}+{52}}
(11) Type: Expression(Integer)
fricas
rsimp(r)
\label{eq12}{{\sqrt{{{e}^{x}}+ 1}}\ {\sqrt{{{e}^{x}}+ 3}}}+{{\left({{e}^{x}}- 7 \right)}\ {\sqrt{{{e}^{x}}+ 1}}}
(12) Type: Union(Expression(Integer),...)