In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include
5 − 2 5 , {\displaystyle {\sqrt {5-2{\sqrt {5}}\ }},}
which arises in discussing the regular pentagon, and more complicated ones such as
2 + 3 + 4 3 3 . {\displaystyle {\sqrt[{3}]{2+{\sqrt {3}}+{\sqrt[{3}]{4}}\ }}.}
Denesting Some nested radicals can be rewritten in a form that is not nested. For example,
3 + 2 2 = 1 + 2 , {\displaystyle {\sqrt {3+2{\sqrt {2}}}}=1+{\sqrt {2}}\,,}
2 3 − 1 3 = 1 − 2 3 + 4 3 9 3 . {\displaystyle {\sqrt[{3}]{{\sqrt[{3}]{2}}-1}}={\frac {1-{\sqrt[{3}]{2}}+{\sqrt[{3}]{4}}}{\sqrt[{3}]{9}}}\,.}
Another simple example,
2 3 = 2 6 {\displaystyle {\sqrt[{3}]{\sqrt {2}}}={\sqrt[{6}]{2}}}
Rewriting a nested radical in this way is called denesting. This is not always possible, and, even when possible, it is often difficult.
Two nested square roots In the case of two nested square roots, the following theorem completely solves the problem of denesting. In this section the use of the radical sign {\displaystyle \textstyle {\sqrt {\;^{\;}}}} denotes, as usual, the positive square root of the radicand, which is supposed to be a positive real number, One has two nested square roots with an expression of the form
α + β r γ + δ r , {\displaystyle {\sqrt {\frac {\alpha +\beta {\sqrt {r}}}{\gamma +\delta {\sqrt {r}}}}},}
where all variables denote rational numbers. By rationalizing the fraction, one gets an expression of the form
a ± b r , {\displaystyle {\sqrt {a\pm b{\sqrt {r}}}},}
where a , b , r {\displaystyle a,b,r} are rational numbers and b > 0 {\displaystyle b>0} . By putting c = r b 2 {\displaystyle c=rb^{2}} , the problem is reduced to denest an expression of the form
a ± c . {\displaystyle {\sqrt {a\pm {\sqrt {c}}}}.}
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