For equations in which the variable(s) is/are under a radical.

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### How to find derivative in radical function?

I need to find derivatives of following functions: $$\frac32x^\frac32-\frac{2x^2}{3}$$ $$-2\sqrt{x}-\frac{-2}{\sqrt{x}}$$ So starting from first one, I have tried to first simplify the fractions to ...
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Question: $$\frac{x}{\sqrt{x^2+1}} = x^4 - x$$ I tried: $$\rightarrow \frac{1}{\sqrt{x^2+1}} = x^ 3 - 1$$ $$\to\frac{\sqrt{x^2 + 1}+1}{\sqrt{x^2+1}} = x^3$$ Now rationalising it $$\to \frac{x^2 +1-1}{... 3 votes 2 answers 321 views ### Understanding Cardano's Formula In deriving his formula, Cardano arrives at the equation y^3+py+q=0. By substituting y=\sqrt{u}+\sqrt{v}, he gets the equation (u+v+q)+(\sqrt{u}\sqrt{v})(3\sqrt{u} \sqrt{v} +p)=... 3 votes 5 answers 340 views ### Solving (x+1)\sqrt{2(x^2 + 1)} + \sqrt{6x^2 + 18x +12} = \frac{3x^2 + 7x + 10}{2} Today, I came across this problem.$$(x+1)\sqrt{2(x^2 + 1)} + \sqrt{6x^2 + 18x +12} = \dfrac{3x^2 + 7x + 10}{2}$$We are asked to find the possible values of x satisfying this equation. The first ... 0 votes 0 answers 53 views ### Expressing the solution to a rational expression in radical form I'm trying to find the solution to this equation$$ -\frac{3}{r} + \frac{8}{r^3} = \frac{\sqrt{2}-1}{2} $$but I haven't been able to find a solution in radical form. Although I've found the solutions ... 1 vote 2 answers 871 views ### Why can't a negative base be raised to a non-integer exponent? Why can't we do this? Example: (-1) ^ {1/3} Math definitions are based on a definite logic. What is the logic here? Can you give me some examples where it violates the equation? I'm just a high ... 5 votes 3 answers 674 views ### If \sqrt{a}\sqrt{b}=\sqrt{ab} only holds for positive real a & b, then why can we say \sqrt{-a}=\sqrt{-1\cdot a}=\sqrt{-1}\sqrt{a}=i\sqrt{a}? I am a little bit bummed that I have this question as I'm sure it has been asked before (I couldn't find the answer) but... If \sqrt{a}\sqrt{b} = \sqrt{ab} is only true for positive reals a and ... 7 votes 4 answers 271 views ### Solve for x in \sqrt{x+2\sqrt{x+2\sqrt{x+2\sqrt{3x}}}} = x Solve for x:$$\sqrt{x+2\sqrt{x+2\sqrt{x+2\sqrt{3x}}}} = x$$I tried to substitute y=x+2 and then I try to solve the equation by again and again squaring. Then I got equation,$$(y-2)(3y^{14}-(y-... 307 views

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### Solve the equation $\sqrt{x^2-1}=(x+5)\sqrt{\frac{x+1}{x-1}}$

Solve the equation $$\sqrt{x^2-1}=(x+5)\sqrt{\dfrac{x+1}{x-1}}.$$ I think that radical equations can be solved by determining the domain (range) of the variable and at the end the substitution won't ...
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### Solve $\frac{7}{x+\sqrt{x+5}}+\frac{7}{x-\sqrt{x+5}}=8$

Solve the equation: $$\dfrac{7}{x+\sqrt{x+5}}+\dfrac{7}{x-\sqrt{x+5}}=8.$$ I am not sure how to approach the problem. Should we first determine the domain? I think we can also check for every value we ...
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### Zero set of nested radicals

My question deals with a function on $\mathbb{R}^n$ that consists of nested radicals and polynomial functions. I'm not even sure how to properly formulate this question, i.e. precisely what class of ...
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