Questions tagged [nested-radicals]
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.
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In general, nested and unnested radical extensions have distinct Galois groups?
Say i have two distinct Galois field extensions $E_1/F$ and $E_2/F$, $F$ a zero characteristic field, such that the two following cases hold:
Case 1
The extension forms by ajoining the elements $\...
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Properties of Nth roots and fractional powers
Context: I'm programming an arbitrary precision math library and created some weird algorithms to calculate a number raised to non-integer powers due to optimizations.
From my understanding, raising ...
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Cosine as nested roots
I have been playing around with circles lately, and I have found an interesting limited relationship between prime factors and cosine. Have the form of:
$$\cos{\left(2\pi\frac{1}{p}\right)}$$
And that ...
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Evaluating $ \sqrt{1+4\sqrt{1+9\sqrt{1+16\sqrt{1+\cdots}}}} $
The remarkable Ramanujan nested radical is
$ \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}=3 $
What can be said about
$ \sqrt{1+4\sqrt{1+9\sqrt{1+16\sqrt{1+\cdots}}}} $ ?
With Mathematica I found that ...
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Crazy integral with nested radicals and inverse sines
Recently a friend who is writing a book on integrals added this problem to his book:
$$\int_{0}^{1}\arcsin{\sqrt{1-\sqrt{x}}}\ dx=\frac{3\pi}{16}$$
After a while, when trying to generalize, I was able ...
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An infinite nested radical [closed]
Can anyone help me in finding a closed form of the infinite nested radical here
$$\left({\sqrt {4+\sqrt {4+\sqrt {4-\sqrt {4+\sqrt {4+\sqrt {4- ......\infty}}}}}}}\right)$$
The signs are as "+,+,-...
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Solvability by radicals, but you don't get to choose the roots
It is well-known that a polynomial equation $P(X)=0$ over a field $K$ is solvable by radicals if and only its Galois group is solvable. Here, "solvabile by radicals" is taken to mean that ...
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Pell numbers $a_n$, prove that $\sqrt{2a_{2n-1}\pm\sqrt{2}}\in\Bbb Q[e^{i\pi/8}]$
$$
a_0 = 0, \quad a_1 = 1; \quad \text{for } n > 1, \quad a_n = 2 \cdot a_{n-1} + a_{n-2}
$$
is the sequence 1, 2, 5, 12, 29, …
$2a_{2n-1}$
is the sequence 2, 10, 58, 338, 1970, …
I try to prove
$$\...
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Infinite nested radical and fractions [duplicate]
By dealing with infinite radicals
I'm pretty sure that we always choose the positive roots,
as an example:
$\displaystyle\sqrt{1+\displaystyle\sqrt{1+...} }$
$\displaystyle\sqrt{1+x}=x$
$x^2=x+1$
This ...
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If $\sqrt[n]{x + \sqrt{y}} + \sqrt[n]{x - \sqrt{y}}$ is an integer, can we always denest $\sqrt[n]{x + \sqrt{y}}$ as $(p + \sqrt{q})/2$?
I'll use $\sqrt[k]{\cdot}$ to denote the principal real $k$-th root of a real-valued input, i.e. the maximum real $k$-th root if it exists and undefined otherwise.
Consider integers $n, x, y, z > 0$...
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Infinitely nested radicals under fraction
Good morning everyone, I encountered a calculation procedure (structural dimensioning) that proceeds by trial and error in a recursive manner, which converges to a value. From various steps I arrived ...
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Proving the Value of a Unique Infinitely Nested Radical
Math Stack Exchange!
I am trying to figure out how to find value of the infinitely nested radical
$$x = \sqrt{2^0 + \sqrt{2^2 + \sqrt{2^4 + \sqrt{2^8 + \ldots}}}}$$
I have already established that ...
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Is this method of solving finite nested square roots of 2 via Gray code correct?
The "Nested square roots of 2" section of the Wikipedia entry "Nested radical" (https://en.m.wikipedia.org/wiki/Nested_radical) describes some properties of finite nested square ...
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How do I simplify $\sqrt{\frac{1-\frac{\sqrt5 }5}2}$?
I've been stuck on simplifying this nested radical. I've included a snapshot of the problem and solution that is in the trigonometry book that I am studying.
I've omitted the actual trig problem and ...
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Infinitely nested radical $\sqrt{1^2+\sqrt{2^2+\sqrt{4^2+\sqrt{8^2+\sqrt{16^2+\sqrt{32^2+\cdots}}}}}}$
Recently, I saw this intriguing radical, which is infinitely nested. I tried to de-nest it but could not due to the square of terms in a geometric. By the technique of partial terms (heuristically), ...
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