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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### Find the two complex numbers whose sum is $5 - i$ and product is $8+i$

I have tried to solve this problem, but when I write the equations I end up with: $$a + c = 5$$ $$b + d = -1$$ $$ab - cd = 8$$ $$ad + cb = 1$$ But substituting $a$ and $b$ get 2 equations I can't be ...
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Consider the function $f$ defined as the limit of the functions $$f_0(x)=\sqrt{x}$$ $$f_1(x)=\sqrt{x+\sqrt{x+1}}$$ $$f_2(x)=\sqrt{x+\sqrt{x+1+\sqrt{x+2}}}$$ $$...$$ so that $f(x)$ is defined iff $f_n(... 0answers 15 views ### An example of nested radical and power tower .$e^{-1}=(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{\cdots}}}}}}$I want to share with you some of my last work: $$e^{-1}=(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{\cdots}}}}}}$$ It's easy to solve using logarithm but I would like ... 1answer 60 views ### Conjecture on a functional equality and nested radical of Ramanujan :$3=\sqrt{1+f(2)\sqrt{1+f(3)\sqrt{1+\cdots}}}$Hi i was wondering something about the great Ramanujan : I think moreover I'm not the only one who propose this kind of problem (so if you have a link related to this subject). We have : $$3=\sqrt{1+... 0answers 24 views ### Almost integer with nested radicals and power tower . playing with power tower and nested radicals I get : Prove that Let a_1=\sqrt{2} ,a_2=\sqrt{2}^{\sqrt{2}},a_3=\sqrt{2}^{\sqrt{2}^{\sqrt{2}}},a_4=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}... 0answers 89 views ### How to evaluate the integral \int_{-1}^{0}\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{\cdots\sqrt{1+x}}}}}dx=? We have :$$\int_{-1}^{0}\sqrt{1+\sqrt{1+\sqrt{1+x}}}dx=\frac{8}{315}\sqrt{2}\Big(16+\sqrt{233+317\sqrt{2}}\Big)$$We are lucky because this integral have an anti-derivative like here. More ... 1answer 39 views ### On proving an infinite-nested radical I was playing around with square roots and I noticed that the number 1 can be seemingly expressed as an infinite nested radical with an easy pattern. I then noticed that if this is true, this would ... 2answers 54 views ### How to solve 10\sqrt{10\sqrt{10\sqrt{10…}}}? [closed] How to solve 10\sqrt{10\sqrt{10\sqrt{10...}}}? I tried to solve this problem by letting x=10\sqrt{10\sqrt{10\sqrt{10...}}} to observe the pattern. Based on the pattern, the result is ... 0answers 32 views ### Abel–Ruffini theorem, connection to nested radicals I hope someone can help me with understanding the Abel–Ruffini theorem, especially the connection to nested radicals. I do know the proof based on Galois theory, but I wondered if this text which is ... 1answer 37 views ### Complex nested radicals {\Re}\Big(\sqrt{1+\frac{i}{2}\sqrt{1+\frac{i}{2^2}\sqrt{1+\frac{i}{2^3}\sqrt{1+\frac{i}{2^4}\sqrt{\cdots}}}}}\Big)=1 A Last question on nested radicals but this time with complex value :$${\Re}\Big(\sqrt{1+\frac{i}{2}\sqrt{1+\frac{i}{2^2}\sqrt{1+\frac{i}{2^3}\sqrt{1+\frac{i}{2^4}\sqrt{\cdots}}}}}\Big)=1$$I have ... 0answers 25 views ### Algebra point of view on Tribonacci constant with nested radical Trying to find an expression in term of nested radical for the tribonacci constant we get a messy result starting with :$$S=\sqrt{1+S+\frac{1}{S}}$$I have tried another expression like :$$S=\... 0answers 41 views ### Special power tower$x^{\sqrt{x+x^{\sqrt{x+x^{\sqrt{\cdots}}}}}}$and generalized Lambert's function I'm interested by the following "nested radical-power tower" we have : $$x^{\sqrt{x+x^{\sqrt{x+x^{\sqrt{\cdots}}}}}}=S$$ My try : We have taking logarithm on both side : $$\ln(S)=\sqrt{x+S}\ln(x)$$ ... 0answers 44 views ### Inequality of the April month$a^{\sqrt{b+b^{\sqrt{2b}}}}+b^{\sqrt{a+a^{\sqrt{2a}}}}\leq 1$the main idea was to create an inequality based on the well know inequality of Vasile Cirtoaje and add some nested radicals it gives : Let$a,b>0$such that$a+b=1$then we have : $$a^{... 1answer 53 views ### Find m^3 if m=\sqrt{a + \frac{a+8}{3}\sqrt{\frac{a-1}{3}}} + \sqrt{a - \frac{a+8}{3}\sqrt\frac{a-1}{3}} Please help me solve this question in a easy way:$$ \sqrt{a + \frac{a+8}{3}\sqrt{\frac{a-1}{3}}} + \sqrt{a - \frac{a+8}{3}\sqrt\frac{a-1}{3}} = m $$Find m^3. (The answer is 8.) I ... 3answers 102 views ### Showing \frac34=\sqrt{1-\frac{1}{2}\sqrt{1-\frac{1}{4}\sqrt{1-\frac{1}{8}\sqrt{\cdots}}}}$$\frac34=\sqrt{1-\frac{1}{2}\sqrt{1-\frac{1}{4}\sqrt{1-\frac{1}{8}\sqrt{\cdots}}}}$$We cannot use the Herschfeld's Convergence Theorem because you can see that it's negative. I have tried some ... 3answers 79 views ### Solutions of the equation x=\sqrt{2 + \sqrt{2 +\sqrt 2+…}} I have seen the equation x=\sqrt{2 + \sqrt{2 +\sqrt 2+.....}} in many places and the answer is x=2 which is obtained by making the substitution for x inside the radical sign i.e x=\sqrt{2+x} ... 1answer 52 views ### Proving that there exists n,a \in \mathbb{N} such that n = \sqrt{a+\sqrt{a+\sqrt{a+\sqrt{a+…}}}} for all n \in \mathbb{N} I suspect that there are infinite solutions of n,a \in \mathbb{N} for all n \in \mathbb{N}, n>1 such that$$n = \sqrt{a+\sqrt{a+\sqrt{a+\sqrt{a+...}}}}$$Some solutions include, for (n,a): ... 1answer 33 views ### How to simplify fractions involving surds? \frac{(\sqrt{θ^2+1})(2θ)-(θ^2-1)(\frac{θ}{\sqrt{θ^2+1}})}{θ^2+1} I came across this example of the Quotient Rule and can't understand how it was simplified from step 2 to 3. How were the roots turned into the power of 3/2? Find \frac{du}{d\theta} when u = ... 2answers 74 views ### Determine whether \sqrt[n]{1+\sqrt[n]{2+\sqrt[n]{3+\cdots+\sqrt[n]{n}}}} diverges or not Determine whether \{a_n\} is convergent or not, where$$a_n:=\sqrt[n]{1+\sqrt[n]{2+\sqrt[n]{3+\cdots+\sqrt[n]{n}}}}.$$At least, we can obtain$$\sqrt[n]{1+\sqrt[n]{2+\sqrt[n]{3+\cdots+\sqrt[n]{n}}... 0answers 45 views ### Does this have repeated operation have a name? For some$k\in \mathbb{R}$taking$\sqrt{\sqrt{\dotsb\sqrt{k}}}$with$N$radicals, then taking$N\to \infty$, will give$1$. Does this series of operations have a name? 2answers 178 views ### Evaluating$\sqrt{9-5\sqrt{3-\sqrt{9-5\sqrt{3-\sqrt{9-\cdots}}}}}$. I was wondering if it was possible to evaluate $$\sqrt{9-5\sqrt{3-\sqrt{9-5\sqrt{3-\sqrt{9-5\sqrt{3-\sqrt{9-\cdots}}}}}}}$$ I let the expression equal$x>0$and wrote $$x=\sqrt{9-5\sqrt{3-x}}$$ ... 2answers 140 views ###$a_n=\sqrt{2-a_{n-1}},a_1=\sqrt{2}$Given $$a_n=\sqrt{2-a_{n-1}},a_1=\sqrt{2}$$ I calculated$a_1$to$a_5\sqrt{2}, \sqrt{2-\sqrt{2}}, \sqrt{2-\sqrt{2-\sqrt{2}}}, \\ \sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}, \\ \sqrt{2-\sqrt{2-\sqrt{2-\... 0answers 107 views ### Value of \sqrt{\sum_{k=1}^{1}k+\sqrt{\sum_{k=1}^{2}k+\sqrt{\sum_{k=1}^{3}k…}}} Note: I would like to add this is very similar to this problem posted a couple months ago, which is the inspiration for this question, was proven to converge in one of the answers, and I believe a ... 2answers 114 views ### Minimal polynomial of \sqrt{3+2\sqrt{3}} Let a=\sqrt{3+2\sqrt{3}}. Then \begin{align*} &a=\sqrt{3+2\sqrt{3}}\\ &\implies a^2=3+2\sqrt{3}\\ &\implies a^2-3 = 2\sqrt{3}\\ &\implies (a^2-3)^2 = 4\cdot 3=12\\ &\implies (a^2-... 1answer 87 views ### Denesting \sqrt{20+ \sqrt{96}+\sqrt{12}} into four possible radicals Denesting \sqrt{20+ \sqrt{96}+\sqrt{12}} into possible radicals. This is an answer to an obscure closed question, here on the site. While there is an answer posted, it isn't the complete solution. 3answers 128 views ### Prove that \sqrt{18 + \sqrt{325}} + \sqrt{18 - \sqrt{325}} = 3 without using Cardano's formula. (Hint, what is (3\pm \sqrt{13})^3 Prove that \sqrt{18 + \sqrt{325}} + \sqrt{18 - \sqrt{325}} = 3 without using Cardano's formula. (Hint, what is (3\pm \sqrt{13})^3 I have that(3 + \sqrt{13})^3 = 144 + 40 \sqrt{13} $$... 1answer 60 views ### Infinite complex nested radical and it's complex conjugate. Today I want to play with i the imaginary unit I have this :$$\overline{\sqrt{1+i\sqrt{1+i^2\sqrt{1+i^3\sqrt{1+i^4\sqrt{\cdots}}}}}}=\sqrt{1+\frac{1}{i}\sqrt{1+\frac{1}{i^2}\sqrt{1+\frac{1}{i^3}\... 3answers 108 views ### Show\frac{\sqrt{\sqrt8-\sqrt{\sqrt2+1}\;}}{\sqrt{\sqrt8+\sqrt{\sqrt2-1}\;} -\sqrt{\sqrt8-\sqrt{\sqrt2-1}\;}}=\frac1{\sqrt2}$Days ago, I tried to demonstrate this equality, reducing radicals, multiplying by the conjugate of the denominator, etc. But, I did not reach anything similar to the right side. $$\frac{\sqrt{\sqrt[... 1answer 108 views ### How to prove that \sqrt{1+\frac{1}{2}\sqrt{1+\frac{1}{3}\sqrt{1+\frac{1}{4}\sqrt{\cdots}}}}<\sqrt 2 It's an estimation that I find interesting :$$\sqrt{1+\frac{1}{2}\sqrt{1+\frac{1}{3}\sqrt{1+\frac{1}{4}\sqrt{1+\frac{1}{5}\sqrt{\cdots}}}}}<\sqrt[\leftroot{-0}\uproot{0}3]{2}$$I think to ... 1answer 120 views ### Nested radical and rationality I did not found it on the forum so :$$S=\sqrt{1+\frac{1}{2}\sqrt{1+\frac{1}{2^2}\sqrt{1+\frac{1}{2^3}\sqrt{1+\frac{1}{2^4}\sqrt{\cdots}}}}}=1.25$$I try to denested the radical but I'm really stuck ... 1answer 48 views ### Nested radical of Ramanujan and cyclic inequality Inspired by a nested radical of Ramanujan I propose this : Let a,b,c\in (2\sqrt[4\,]{5},6-2\sqrt[4\,]{5}) such that a+b+c=9 then we have :$$\sqrt[4\,]{\frac{a+2\sqrt[4\,]{5}}{a-2\sqrt[4\,]{5}... 1answer 46 views ### Denest$\sqrt{3+\sin\frac{\pi}{4}-4\cos\frac{\pi}{8}}\$ in terms of possible trigonometric functions.
As the title suggests, denest; $$\sqrt{3+\sin\frac{\pi}{4}-4\cos\frac{\pi}{8}}$$ in terms of possible trigonometric functions.