Nested radical

In algebra, nested radicals are radical expressions that contain another radical expression. Examples include

sqrt{5-2sqrt{5} }

which arises in discussing the regular pentagon,

sqrt{5+2sqrt{6} },

or more complicated ones such as

sqrt[3]{2+sqrt{3}+sqrt[3]{4} }.

Denesting these radicals is generally considered a difficult problem. A special class of nested radical can be denested by assuming it denests into a sum of two surds:

sqrt{a+b sqrt{c} } = sqrt{d}+sqrt{e},

a+b sqrt{c} = d + e + 2 sqrt{de};

this can be solved by the quadratic formula and by setting rational and irrational parts on both sides of the equation equal to each other.

In some cases, higher-power radicals may be needed to denest certain classes of nested radicals.

Infinitely nested radicals

Square roots

Under certain conditions infinitely nested square roots such as

x = sqrt{2+sqrt{2+sqrt{2+sqrt{2+cdots}}}}

represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign, which gives the equation

x = sqrt{2+x}.

If we solve this equation, we find that x = 2 (the second solution x = −1 doesn't apply, under the convention that the positive square root is meant). This approach can also be used to show that generally, if n > 0, then:

sqrt{n+sqrt{n+sqrt{n+sqrt{n+cdots}}}} = frac{1 + sqrt {1+4n}}{2}.

The same procedure also works to get

sqrt{n-sqrt{n-sqrt{n-sqrt{n-cdots}}}} = frac{-1 + sqrt {1+4n}}{2}.

This method will give a rational x value for all values of n such that

{n} = {x^2} + {x}. ,

Cube roots

In certain cases, infinitely nested cube roots such as

x = sqrt[3]{6+sqrt[3]{6+sqrt[3]{6+sqrt[3]{6+cdots}}}}

can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are left with the equation

x = sqrt[3]{6+x}.

If we solve this equation, we find that x = 2. More generally, we find that

sqrt[3]{n+sqrt[3]{n+sqrt[3]{n+sqrt[3]{n+cdots}}}} is the real root of the equation x^3-x-n=0 ,! for all n where n > 0.

The same procedure also works to get

sqrt[3]{n-sqrt[3]{n-sqrt[3]{n-sqrt[3]{n-cdots}}}} as the real root of the equation x^3+x-n=0 ,! for all n and x where n > 0 and |x| ≥ 1.


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