For questions involving radical of numbers or radical of expressions (i.e. numbers/expressions raised to the power of a fraction).

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### Difficult expressions that equate to a rational number

Currently I'm learning to use Cardano's Method to solve cubic equations and one of the interesting things is that, even for an equation with simple solutions, you may still have to generate some ...
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### Prove that $\frac{1}{\sqrt{bc+a}}+\frac{1}{\sqrt{ca+b}}+\frac{1}{\sqrt{ab+c}}\ge \frac{\sqrt{6}}{\sqrt{ab+bc+ca}}$

For all $a,b,c$ are positive real numbers which has a sum of 3. Prove that: $$\frac{1}{\sqrt{bc+a}}+\frac{1}{\sqrt{ca+b}}+\frac{1}{\sqrt{ab+c}}\ge \frac{\sqrt{6}}{\sqrt{ab+bc+ca}}$$ The problem was ...
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### Denesting Method for $\sqrt[3]{-10+9\sqrt{-3}}$ (Without Solving Another Cubic Equation)?

To solve the cubic equation $$x^3-2x^2-x+2=0$$such an answer is obtained: $$\sqrt[3]{-10+9\sqrt{-3}}$$ which can not be easily denested manually. For denesting that radical, I had to solve another ...
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### Can a simple neural network model(MLP) predict or fit the square root of x?

According to the description in the Can AI Predict What Will Happen? section of the article Can AI Solve Science? on stephenwolfram.com, ... there are "no model-less models": different ...
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### Other methods of solving this question: finding $2y^4-8y^3-5y^2+26y-28$ for $y=1+\sqrt2+\sqrt3$ [closed]

If $y=1+\sqrt2+\sqrt3$ then find the value of $$2y^4-8y^3-5y^2+26y-28$$ There can be many ways to do this question. I would like to know the shorter approaches for this question that are clever to ...
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### Simplifying the conversion of a rational exponent to radical form

I've learned that $x^ \frac nm$ should be turned into radical form like $\sqrt[m]{x^n}$. Therefore $10^ \frac32$ should be turned into radical form like $\sqrt {10^3}$. What I don't understand is ...
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### What does $\cos^2x$ mean? (only the 2 is an exponent) [closed]

What does the expression $\cos^2x$ mean? For context, the 2 is the exponent, and the $x$ is not in superscript. $$I=I_0\cos^2\theta$$ The above equation shows the context I'm looking for. If you ...
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### Help with Evaluating a Definite Integral Involving Nested Radicals

I'm working on a calculus problem and need help solving the following definite integral: I'm struggling to simplify the integrand or find a substitution that makes the integral easier to evaluate. ...
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### Calculating the fixed-point representation of (1 - √0.5) to arbitrary levels of precision

I have a constant 0.29289321881345247559915563789515..., which can be calculated using the equation (1 - √0.5) and then ...
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### Funny lattice equivalence between additive and subtractive extended square roots

I am interested in the behavior of the "extended square root", such as: $$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt2}}} ...$$ $$\sqrt{n\pm\sqrt{n\pm\sqrt{n\pm\sqrt{n\pm}}}} ...$$ I've noticed after ...
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### Is this method I made, (Babylonian Approximix) to find square roots with an approximate answer accurate and useful?

Step 1: create a guess on the square root that is reasonably close when you multiply it by itself. Step 2: Divide the guess number by the number you are trying to get the square root of. Step 3. ...
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### Prove that $\sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$ is irrational [duplicate]

Prove that $\sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$ is irrational I tried to let $x = \sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}$, and assume for the sake of contradiction that $x$ is rational. ...
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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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### How to elegantly solve $\left(x^2-\frac1{x^2}\right)^2+2\left(x+\frac1{x}\right)^2 = 2024$?
Find all values of $x$ such that $$\left(x^2-\frac{1}{x^2}\right)^2\:+\:2\,\left(x+\frac{1}{x}\right)^2 \:=\: 2024$$ I arrived at the solutions $\,x=\pm \sqrt{22 \pm \sqrt{483}}$, \$\,\pm \sqrt{-23 \...