which arises in discussing the regular pentagon;
or more complicated ones such as:
Denesting nested radicals
Some nested radicals can be rewritten in a form that is not nested. For example,
Rewriting a nested radical in this way is called denesting. This process is generally considered a difficult problem, although a special class of nested radical can be denested by assuming it denests into a sum of two surds:
Squaring both sides of this equation yields:
This can be solved by equating coefficients of like terms—setting rational and irrational parts on both sides of the equation equal to each other. The solutions for e and d can be obtained by first equating the rational parts:
For the irrational parts note that
and squaring both sides yields
By plugging in a − e for d one obtains
Rearranging terms will give a quadratic equation which can be solved for e using the quadratic formula:
Since a = d+e, the solution d is the algebraic conjugate of e. If we set
In some cases, higher-power radicals may be needed to denest the nested radical.
Some identities of Ramanujan
Other odd-looking radicals inspired by Ramanujan include:
In the solution of the cubic equation
whose general solution for one of the roots is
here the first cube root is defined to be any specific cube root of the radicand, and the second cube root is defined to be the complex conjugate of the first one. The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, we have the casus irreducibilis, in which all three real solutions are written in terms of cube roots of complex numbers. On the other hand, consider the equation
which has the rational solutions 1, 2, and —3. The general solution formula given above gives the solutions
For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet it is reducible (even though not obviously so) to one of the solutions 1, 2, or –3.
Infinitely nested radicals
Under certain conditions infinitely nested square roots such as
represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign, which gives the equation
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:
and is the real root of the equation x2 − x − n = 0. For n = 1, this root is the golden ratio φ, approximately equal to 1.618. The same procedure also works to get that
and is the real root of the equation x2 + x − n = 0. For n = 1, this root is the reciprocal of the golden ratio Φ, which is equal to φ − 1. This method will give a rational x value for all values of n such that
Ramanujan posed this problem to the 'Journal of Indian Mathematical Society':
This can be solved by noting a more general formulation:
Setting this to F(x) and squaring both sides gives us:
Which can be simplified to:
It can then be shown that:
So, setting a =0, n = 1, and x = 2:
Ramanujan stated this radical in his lost notebook
(The repeating pattern of the signs is
In Viète's expression for pi
In certain cases, infinitely nested cube roots such as
can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are left with the equation
If we solve this equation, we find that x = 2. More generally, we find that
is the real root of the equation x3 − x − n = 0 for all n > 0. For n = 1, this root is the plastic number ρ, approximately equal to 1.3247.
The same procedure also works to get
as the real root of the equation x3 + x − n = 0 for all n and x where n > 0 and |x| ≥ 1.
- "A note on 'Zippel Denesting'", Susan Landau, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.35.5512&rep=rep1&type=pdf
- "RADICALS AND UNITS IN RAMANUJAN’S WORK", Susan Landau, http://www.math.uiuc.edu/~berndt/articles/radicals.ps
- Landau, Susan (1992). "Simplification of Nested Radicals". Journal of Computation (SIAM) 21: 85–110. doi:10.1109/SFCS.1989.63496. CiteSeerX: 10
.1 .1 .34 .2003.
- Landau, Susan (1994). "How to Tangle with a Nested Radical". Mathematical Intelligencer 16: 49–55. doi:10.1007/bf03024284.
- Decreasing the Nesting Depth of Expressions Involving Square Roots
- Simplifying Square Roots of Square Roots
- Weisstein, Eric W., "Square Root", MathWorld.
- Weisstein, Eric W., "Nested Radical", MathWorld.