Nested radical

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In algebra, nested radicals are radical expressions that have another radical expression nested inside a radical. Examples include

$\sqrt{5-2\sqrt{5}\ }$

which arises in discussing the regular pentagon,

$\sqrt{5+2\sqrt{6}\ },$

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.

1 Infinitely nested radicals
- 1.1 Square roots
- 1.2 Cube roots
2 References

[edit] Infinitely nested radicals

[edit] 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:

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