help links subscribe changes refresh edit

	FriCAS Library Examples
PolynomialOverFiniteField ←	↑	→ Sqrt3Demo

Root denesting

Edit detail for Root denesting revision 1 of 1

	1
Editor: test1 Time: 2024/01/12 14:31:01 GMT+0
Note:

changed:
-
FriCAS can denest many nested roots.  This is implemented in an extra package so
befor use we need to expose it:
\begin{axiom}
)expose RootSimplification
\end{axiom}

Now we can try
\begin{axiom}
r := sqrt(102*sqrt(7) + 272)
rsimp(r)
r := sqrt(66000162*1000003^(1/2) + 11000159000324)
rsimp(r)
r := sqrt(((-4030008370)*700001^(1/2)+12090025110)*1000003^(1/2)+((-78000162)*700001^(1/2)+312335685800846295))
rsimp(r)
\end{axiom}

We can denest roots of higher degree:
\begin{axiom}
r := (((-6670600520850)*7^(1/2)+73781123539185)*13^(1/2)+((-77245760121201)*7^(1/2)+89932095680661))^(1/7)
rsimp(r)
\end{axiom}

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:
\begin{axiom}
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)
rsimp(r)
\end{axiom}

Denesting is not limited to numbers, it works for algebraic functions and more general expressions:
\begin{axiom}
)set output tex off
)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)
rsimp(r)
)set output algebra off
)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)
rsimp(r)
\end{axiom}

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)

Type: AlgebraicNumber?

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)

Type: AlgebraicNumber?

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)

Type: AlgebraicNumber?

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)

Type: AlgebraicNumber?

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)

Type: AlgebraicNumber?

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),...)