Questions tagged [radicals]
For questions involving radical of numbers or radical of expressions (i.e. numbers/expressions raised to the power of a fraction).
3,685
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Use asymptotic equality to solve limit of difference of functions
I have to compute the following limit
$$\lim_{x\to+\infty}f(x)=\lim_{x\to+\infty}\sqrt{x^2+1}-\sqrt{x^2+x}$$
I have rewritten the function $f(x)$ as
$$f(x)=\frac{x^2+1-x^2-x}{\sqrt{x^2+1}+\sqrt{x^2+x}}...
- 171
1
vote
1
answer
71
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If $\alpha^3-\alpha+1=0$ then $\sqrt{3}\neq a\alpha^2+b\alpha+c$ for $a$, $b$, $c\in \mathbf{Q}$
Im stuck on the following:
Let $f(x) =x^3-x+1$ and let $\alpha$ be a real root of $f$. Prove that $$\sqrt{3} \neq a\alpha^2 +b\alpha +c$$ for any $a$, $b$, $c\in \mathbf{Q}.$
I am preparing for a ...
- 1,003
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0
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45
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What are the solutions for the inequality $x^2 \geq 60$ [duplicate]
I searched it up on wolfram and the result it shows is:
$x \geq 2 \sqrt {15}$ and $x \leq -2 \sqrt {15}$.
What I am doing is:
$x \geq \pm 2\sqrt {15}$
Now, two cases arise,
$x \geq 2\sqrt {15}$ or $...
- 1
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1
answer
46
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Confused about the extraneous root of $\frac{x \sqrt{A^2 - x^2} + x}{x^2 - \sqrt{A^2 - x^2}}$
I have:
$$f(x) = \frac{x \sqrt{A^2 - x^2} + x}{x^2 - \sqrt{A^2 - x^2}}$$
to find the roots:
$$x \sqrt{A^2 - x^2} + x = 0$$
$$\sqrt{A^2 - x^2} + 1 = 0$$
$$\sqrt{A^2 - x^2} = -1$$
$$A^2 - x^2 = 1$$
$$A^...
- 105
8
votes
1
answer
277
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How to prove $\sqrt{2a+3bc}+\sqrt{2b+3ca}+\sqrt{2c+3ab}\ge 3\sqrt{5}$?
If $a,b,c\ge 0: ab+bc+ca=3$ then prove that: $$\sqrt{2a+3bc}+\sqrt{2b+3ca}+\sqrt{2c+3ab}\ge 3\sqrt{5}.$$
I've tried to use AM-GM and Holder without success.
Indeed, by AM-GM for three numbers $$\sqrt{...
- 778
0
votes
2
answers
77
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Tedious rational integral
I have been trying to solve this tedious integral, but I am unable to find a substitution that helps to solve it. $$\int\frac{\sqrt{x^2+x+2-\sqrt{4x^2+4x+4}}}{x\sqrt{x^4+x^3+x^2}}dx$$
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2
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80
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Is it possible to turn $\sqrt\frac12$ into $\sqrt2$? $ \frac{3}{4}\sqrt{\frac{1}{2} } = \frac{3}{8}\sqrt{{2} }$: What is between the left/right sides?
The textbook has the following line:
$$ \frac{3}{4}\sqrt{\frac{1}{2} } = \frac{3}{8}\sqrt{{2} } $$
I don't get what is in-between the left and right sides of the equation.
- 13
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1
answer
75
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Why is $(-3)^\frac{2}{5}$ not real?
Solving some past year school questions, I noticed this simple exponentiation question.
$$
\text{True/False?} \\
((-3)^2)^\frac{2}{5} = ((-3)^\frac{2}{5})^2
$$
At first, this seemed to be right, ...
- 133
1
vote
1
answer
118
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Solving a complex ODE with complex exponents
I have following ODE, with the constraint that $n\neq1$:
$$\bigg(y'\bigg)^{n-1}=ny^{n-1}$$
This is easily solveable in $\mathbb{R}$, however in this case both $y(z)$ and $n$ can be complex.
I would ...
- 85
4
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1
answer
151
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How to integrate $\int \frac{1}{\sqrt{x+1}-\sqrt{x+3}+\sqrt{x+5}}dx$?
How to integrate $$\int \frac{1}{\sqrt{x+1}-\sqrt{x+3}+\sqrt{x+5}}dx$$.
In my school life , I learnt how to integrate $$\int \frac{1}{\sqrt{x+a}-\sqrt{x+b}}dx$$ or $$\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}...
- 1,139
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votes
4
answers
87
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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 ...
- 689
-1
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1
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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 ...
2
votes
1
answer
83
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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 ...
- 778
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votes
7
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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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2
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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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1
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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 ...
- 689
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1
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75
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How to express $y$ from $x^2+y^2>r$?
Consider an inequality in form of circle equation (with zero coordinates):
$$y^2 + x^2 > 1$$
Expressing $y$ step-by-step (maybe I wrong somewhere?):
$$y^2 > 1 - x^2 (1)$$
$$y>\sqrt{(1-x^2)}\...
- 203
2
votes
2
answers
239
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evaluate the limit when x goes to infinity
Evaluate $\displaystyle{\lim_{x \to \infty} (x^3+6x^2+1)^{\frac13}-(x^2}+x+1)^{\frac12}$
I tried writing it as $$\lim_{x \to \infty}\frac{x^3+6x^2+1}{(x^3+6x^2+1)^{\frac23}}-\frac{x^2+x+1}{(x^2+x+1)^{\...
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33
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Definition of simple radical extension.
The definition is: $E$ is a simple radical extension of $F$, if there exists some integer $n$, such that $E=F(a)$, where $a^n\in F$.
Question: let $d$ be the degree $d=[E:F]$, then we can only claim $...
- 159
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0
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124
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Evaluate limit to infinity of roots of $n$ [ $\lim\limits_{n \to \infty} \sqrt n (\sqrt[n]{n}-1)$ ] [duplicate]
Here is the problem:
Evaluate $\lim_{n \to \infty} \sqrt n (\sqrt[n]{n}-1)$
At my disposal:
For $ n \ge 1, n \in \mathbb{N}$ , we have $(\sqrt[n]{n}-1)\le 2/ \sqrt n $
Hence , the upper bound of the ...
- 20.1k
1
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6
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355
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Solving $y+\sqrt{y^2-1}=e^x$ respect to $y$
I have an expression
$$y+\sqrt{y^2-1}=e^x$$
How to find $y$?
I tried it by squaring both sides, after that I also tried to solve by different substitutions like putting $y = \sec (\theta)$, but ...
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1
answer
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Prove $f(a,b,c)\le f\left(\frac{a+b}{2},\frac{a+b}{2},c\right)$
Given non-negative real numbers $a,b,c$ which sum is $2.$ Assume that $c=min\{a,b,c\}.$
Denote $$f(a,b,c)=\frac{\sqrt{2ab+7}+\sqrt{2bc+7}+\sqrt{2ca+7}}{\sqrt{abc+9}}.$$
Prove that $f(a,b,c)\le f(t,t,c)...
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3
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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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votes
3
answers
117
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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 ...
- 689
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1
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28
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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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votes
4
answers
805
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What's wrong with this limit solution?
I was solving this limit:
$$
\lim_{x\rightarrow +\infty} \left( \sqrt{x^2+x+1}-x \right)
$$
And here is my solution:
$$
\lim_{x\rightarrow +\infty} \left( \sqrt{x^2+x+1}-x \right) =\lim_{x\rightarrow ...
- 61
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1
answer
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Finding $\sqrt 2$ in terms of $\sqrt 2 + \sqrt[3]3$
If $\alpha = \sqrt 2 + \sqrt 3$, it is easy to find $\sqrt 2$ in terms of $\alpha$: simply invert $\alpha$ to obtain $\sqrt 3 - \sqrt 2$, hence $\sqrt 2 = (\alpha - 1/\alpha)/2$.
What about if $\beta =...
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Rationalization of $\frac{1}{\sum_{i=1}^{n}\sqrt{a_i}}$ (symmetric formula with multi-index)
Motivation
No particular motivation, I'm just curious to see if an interesting result can emerge from such a seemingly simple problem.
Problem
Find the closed symmetric formula for the rationalization ...
- 3,162
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0
answers
28
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Proving associativity of binary operation
If $S = \{x\in\mathbb{R} \mid x>0\}$ with the binary operation $\star$ given by
$$ x\star y = \sqrt{xy} $$
Then to show S is a group we of course need to show $(S,\star)$ satisfies the group axioms,...
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votes
1
answer
114
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square root of x equals -1
I read that $\sqrt{x} = -1$ has no solution because after we square both sides we get $x = 1,$ which isn't a correct solution. But doesn't writing $-1$ as $i^2$ give the solution $x = i^4$ ?
$$\sqrt{x}...
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votes
3
answers
510
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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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3
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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 ...
- 341
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3
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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 ...
-2
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2
answers
139
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How do you say "ALL the $n$th roots"? [closed]
$\sqrt[2]{1}$ is strictly only $1$, despite the equation $x^{2}=1$ having two solutions: $1$ and $-1$. Same with cube roots; $\sqrt[3]{1}$ is strictly just $1$, despite $\left(-\frac{i\sqrt[2]{3}+1}{2}...
- 356
1
vote
0
answers
67
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Formula for computing rational approximations to square roots
I have been playing around with rational approximations to square roots. I am familiar with https://en.wikipedia.org/wiki/Methods_of_computing_square_roots but did not see it there.
I also did not see ...
1
vote
1
answer
58
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Distributional property of $n$-th root extraction respect to the multiplication
I have seen this exercise in a old book of the 2011. It is this with $a,b\in\Bbb R$.
$$\large \sqrt[10]{a^5b^2}=\sqrt[10]{a^5}\cdot \sqrt[10]{b^2}=\sqrt a \cdot \sqrt[5]{|b|}. \tag 1$$
For my opinion ...
- 6,919
1
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2
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Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$
Let $a$, $b$ and $c$ be nonzero natural numbers. Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$.
My ideas
For those numebrs to ...
- 341
3
votes
2
answers
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Proof of $(\sqrt[n]{a})^n=(\sqrt[n]{a^n})$
Definition: Given a natural number greater than or equal to $1$ and a real number $a\geq 0$, we call the n-th root of $a$ and write $\sqrt[n]{a}$ that nonnegative real number $b$ such that $b^n=a$, i....
- 6,919
0
votes
1
answer
105
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A one word name for "taking a root" operation?
We have addition and subtraction, multiplication and division, But what about raising to a power and taking roots? I hear about "exponentiation" recently, a term that I don't recall being ...
- 294
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vote
0
answers
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Does the notion of a "principal square root" lead to inconsistencies among the complex number system?
The square root of a number $y$ is defined as the the number $x$ that, when squared, results in $y$. For example, the square root of $4$ is $+2$ and $-2$ because both of them result in $4$ when ...
1
vote
2
answers
75
views
Convergence of $u_{n} = \frac{1}{n}+\sqrt{u_{n-1}}$ with $u_0\geqslant0$
I'm trying to know for which $u_0\geqslant 0$ the following sequence
$$
\forall n\in \mathbb{N},\qquad u_{n} = \frac{1}{n}+\sqrt{u_{n-1}}
$$
converges. One can see that : if $u_0\leqslant v_0$ then $...
- 3,979
1
vote
2
answers
156
views
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 ...
- 43
6
votes
0
answers
140
views
Find the fourth roots of the following binomial surd: $14+8\sqrt{3}$
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
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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,...
- 11
9
votes
1
answer
227
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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=-\...
- 591
18
votes
6
answers
974
views
What are some obscure radical identities?
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(\...
- 1,239
-1
votes
1
answer
48
views
Simplification of Radicals [closed]
Recently, I learn some basic factorisation techniques, but then there are sometimes where a messy radical will somehow cancel out into a clean one. Here is one I encountered:
$$\frac{3 \sqrt5 - 5}{3 - ...
- 109
2
votes
0
answers
45
views
integral of squares roots over square roots
I'm trying to do some exercises of this site, but I'm stuck at the letter r one. Specifically this part $$
\int(...) -\frac{
\sqrt[\leftroot{10} \uproot{1} 6]{x\sqrt[\leftroot{10} \uproot{3} ] {x^{...
