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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### Finding $\small{\min\limits_{ab+bc+ca=1}\sqrt{a+2}+\sqrt{b+2}+\sqrt{c+2}- \sqrt{2-abc}.}$

Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Find the minimal value of expression $$P=\sqrt{a+2}+\sqrt{b+2}+\sqrt{c+2}- \sqrt{2-abc}.$$ By $a=b=1;c=0$ I get $P=2\sqrt{3}$ so we ...
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### How does $\lambda \sqrt{\frac{1+v/c}{1-v/c}}$ become $\lambda \frac{1+v/c}{\sqrt{1-v^2/c^2}}$? [closed]

I was reading a physics text and came across this equation : $$\large \lambda \sqrt{\frac{1+ \frac vc}{1-\frac vc}} = \lambda \frac{1+\frac vc}{\sqrt{1-\frac{v^2}{c^2}}}$$ I am confused as to how they ...
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### Find the best constant $k$ such that $\frac{\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}}{\sqrt{a+b+c+k\cdot abc}}\le 1+\sqrt{2}$

Problem. Find the minimal $k$ value such that $$\frac{\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}}{\sqrt{a+b+c+k\cdot abc}}\le 1+\sqrt{2}$$ holds for all $a,b,c\ge 0: ab+bc+ca=1.$ I'm not sure my prediction is ...
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### Is enumerating squares via $(a+1)^2 = a^2+(2a+1)$ a new method to find square roots?

There's a Brazilian 11 year old that allegedly developed an original method of finding natural square roots and is being marketed as some sort of genius for it. The method is even being called "...
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### Getting rid of cube roots in the form of (a+b)+(a-b)

So, I've come across a question that asked "Simplify the sum $\sqrt[3]{18+5\sqrt{13}}+\sqrt[3]{18-5\sqrt{13}}$". As I've had little to no experience with these kinds of questions, I would ...
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### Prove $\frac{1}{\sqrt{a+bc}}+\frac{1}{\sqrt{b+ca}}+\frac{1}{\sqrt{c+ab}}\ge \frac{2\sqrt{2}+1}{2},$ when $a+b+c+abc=4.$

Let $a,b,c\ge 0: a+b+c+abc=4$. Prove that$$\frac{1}{\sqrt{a+bc}}+\frac{1}{\sqrt{b+ca}}+\frac{1}{\sqrt{c+ab}}\ge \frac{2\sqrt{2}+1}{2}.$$ I've try to use AM-GM which is very complicated. Indeed, we ...
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### Finding $\small{\min\limits_{a+b+c=2(ab+bc+ca)}\sum_{cyc}\sqrt{\dfrac{1}{ab}+\dfrac{1}{bc}+1}.}$

If $a,b,c>0 : a+b+c=2(ab+bc+ca),$ then find the minimal value$$P=\sqrt{\dfrac{1}{ab}+\dfrac{1}{bc}+1}+\sqrt{\dfrac{1}{bc}+\dfrac{1}{ca}+1}+\sqrt{\dfrac{1}{ca}+\dfrac{1}{ab}+1}.$$ By set $a=b=c=1/2,$...
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### Finding $\small{\max\limits_{ab+bc+ca=1}\sqrt{\frac{1}{a+1}+\frac{1}{b+1}}+\sqrt{\frac{1}{a+1}+\frac{1}{c+1}}+\sqrt{\frac{1}{c+1}+\frac{1}{b+1}}.}$

Let $a,b,c\ge 0: ab+bc+ca=1.$ Find the maximum $$P=\sqrt{\frac{1}{a+1}+\frac{1}{b+1}}+\sqrt{\frac{1}{a+1}+\frac{1}{c+1}}+\sqrt{\frac{1}{c+1}+\frac{1}{b+1}}.$$ By denote some specific value, I think ...
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### Real-base powers and fractional exponent

We suppose that $a\in\Bbb R^{+}_0$ we know that $$a^{\frac{m}{n}}=\sqrt[n]{a^m} \tag 1$$ Is $(1)$ provable or is it a given definition. Many years ago I remember that perhaps there was a proof of such ...
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### How can we show that $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is a rational number only if $a,b,c$ are perfect squares?

Question I saw a lot of problems that assume this: $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is a rational number only if $a,b,c$ are perfect squares. I wonder how can we demonstrate it because I saw a lot of ...
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### Show that among the numbers $\sqrt{\frac{pq-2}{3}}$, $\sqrt{\frac{qr-2}{3}}$ and $\sqrt{\frac{pr-2}{ 3}}$ there is at least one irrational one.

question Let the natural numbers $p$, $q$ and $r$ be greater than $2$. Show that among the numbers $\sqrt{\frac{pq-2}{3}}$, $\sqrt{\frac{qr-2}{3}}$ and $\sqrt{\frac{pr-2}{3}}$ there is at least one ...
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### How do you find the exact value of a logarithm with a radical in the base?

I'm struggling to find a method for evaluating $\log_{5\sqrt2} 50$ (or ${\log50}\over{\log5\sqrt2}$) without using a calculator. When using a calculator, I am given an exact value of 2, but I can't ...
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### is 0.77777778 the same as 0.8 ? ( Square root problem )

I have holes in my math because I didn't pay attention when I was a kid.( so please explain in detail if possible <3 ) While relearning everything I found my self stuck not understanding how this ...
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Find the fourth roots of the following binomial surd: $X=14+8\sqrt{3}$ I attempt to find the square root first: $\sqrt{X}=\sqrt{14+8\sqrt{3}}$ $\sqrt{14+8\sqrt{3}}=\sqrt{x_1}+\sqrt{y_1}$ $(\sqrt{14+8\... • 1,356 2 votes 1 answer 95 views ### Applying the limit definition of a derivative on a radical function$x^{2/3}$I'm trying to find the derivative of the following using the limit definition of a derivative: $$f(x)=x^{2/3}.$$ I know that the derivative of$f(x)$is$\frac23x^{-1/3}$by the power rule, but I can'... 0 votes 1 answer 98 views ### Calculating the exact square root of a complex number with rational components [duplicate] Given a complex number with rational components, I want to check if its square root also has rational components, and if so calculate the value. For example, given$-\frac{119}{225}+\frac{8}{15}i$I ... • 109 1 vote 0 answers 124 views ### Calculating square roots using perfect squares I recently discovered a way to quickly calculate perfect squares (that are close to a prior-known perfect square), then extrapolated from that a method to mentally calculate the square root of numbers,... 9 votes 1 answer 227 views ### Ratio of theta functions as roots of polynomials I was playing with the theta functions with argument$ z = 0  \vartheta_2(q) =\sum_{n=-\infty}^\infty q^{(n+1/2)^2}  \vartheta_3(q) =\sum_{n=-\infty}^\infty q^{n^2}  \vartheta_4(q) =\sum_{n=-\...
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So there are several trigonometric identities, some very well know, such as $\cos(x) = 1 - 2\sin^2(\frac{x}{2})$ and some more obscure like \$\tan(\frac{\theta}{2} + \frac{\pi}{4}) = \sec(\theta)+\tan(\...