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Questions tagged [radicals]

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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0answers
29 views

Rationalizing denominator with any number of radicals

I'm trying to develop a java class for exact algebraic numbers. I've come to a little bit of a roadblock as far as this goes. My question right now is how to rationalize these equations, but no-one I'...
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7answers
196 views

Hard limit involving different order radicals $\lim_{n \to \infty} (\sqrt[3]{n^3+3n^2}-\sqrt{n^2+2n} )$

Please help me to calculate the following limit $$\lim_{n \to \infty} (\sqrt[3]{n^3+3n^2}-\sqrt{n^2+2n} )$$ I factored out $n$ from both radicals but it didn't work and tried to use the identity $a^2-...
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0answers
9 views

Additional components of the family of identities $ \eta $ functions of level 6.

Can these equations be considered as the family of $\eta$ identities functions of level 6? q6_24_64=729*u1^4*u3^20 – 1944*u1^3*u2^5*u3^15*u6 + 1728*u1^2*u2^10*u3^10*u6^2 – 512*u2^20*u6^4; q6_48_108=...
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2answers
183 views

If $x = \frac{\sqrt{111}-1}{2}$, calculate $(2x^{5} + 2x^{4} - 53x^{3} - 57x + 54)^{2004}$.

I already have two solutions for this problem, it is for high school students with an advanced level. I would like to know if there are better or more creative approaches on the problem. Here are my ...
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0answers
31 views

Proving a number is irrational based off of the fact that another is irrational [duplicate]

Here's the question: Use the fact that √6 is irrational to prove that √2 + √3 is irrational. What would be the best method to use here? How would I lay it out using contradiction? Thanks.
6
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1answer
50 views

Powers of complex numbers.

It is known that if $\alpha$ is a complex number, then for example, the equation $x^2 = \alpha$ has $2$ solutions. In general, there are $n$ values for the $n$-th roots of a number. In other words, if ...
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2answers
36 views

Confusing regarding a nested radical equation

For all $a\in\Bbb R$ solve the equation $$\sqrt{x^2+4a^2\sqrt{x+a}}=x+2a$$ It is immediate to see that we got the restriction $x\geqslant-a$ (even though not given I assume that this equation is ...
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1answer
99 views

Solve this equation $\cos{\left(\frac{\pi}{3}-\frac{\pi}{3r}\right)}=\sqrt{\frac{11}{r^2}-2}$

Let $r>0$, solve this equation $$\cos{\left(\dfrac{\pi}{3}-\dfrac{\pi}{3r}\right)}=\sqrt{\dfrac{11}{r^2}-2}$$ I have found that $r=2$ is a solution, as $$LHS=\cos{\dfrac{\pi}{6}}=\dfrac{\sqrt{3}}...
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2answers
37 views

Simplifying fractional surds

I have this fractional surd: $$\frac{5\sqrt{7}+4\sqrt{2}}{3\sqrt{7}+5\sqrt{2}}$$ I can calculate this with a calculator fairly easily obviously but what is the best tactic without one? Thank you!
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2answers
51 views

Is the $\pm$ sign used when finding the root of a negative number?

If $\sqrt{64}$ is equal to $\pm{}8$, is $-64$ equal to $\pm{}8i$, or just $8i$?
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4answers
61 views

Doubt on how to express $\sqrt{x}^2$ on derivative function… which is the right answer?

I was requested to find the derivative of $f(x)=\frac{x}{\sqrt{1-x^2}}$. When I was working on this, I found an expression of the form $\sqrt{1-x^2}^2$, which I translate to $|1-x^2|$. My calculus ...
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5answers
200 views

Using the fact that $\sqrt{n}$ is an irrational number whenever $n$ is not a perfect square, show $\sqrt{3} + \sqrt{7} + \sqrt{21}$ is irrational.

Question: Using the fact that $\sqrt{n}$ is an irrational number whenever $n$ is not a perfect square, show $\sqrt{3} + \sqrt{7} + \sqrt{21}$ is irrational. Following from the question, I tried: ...
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3answers
96 views

Converting an equation based on square roots

So I was wondering how to convert an equation of the form $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+...\sqrt{x_n}+k=0$ into a polynomial equation based on each $x_i$. For example if the equation was $$\sqrt{...
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1answer
31 views

Exact formula of the roots of a polynomial

I'm looking for a closed formula given one (or all) root of a polynomial $P=aX^4 +bX^3+cX^2+dX+e$. I'm not interested in the efficiency of such a formula. On the contrary, I would like to show my ...
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0answers
38 views

Infinite sum of alternating reciprocals of square roots? $\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt{n}}$ [duplicate]

I'm trying to find the sum of alternating reciprocals of square roots: $$ \sum_{n=1}^\infty\frac{(-1)^n}{\sqrt{n}}=-\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}-\dots $$ ...
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3answers
46 views

Square root of square as 2/2 power

From school I know that $\sqrt{x^2} = |x|$. But if we rewrite the above equation in another way $\sqrt{x^2} = (x^2)^{\frac{1}{2}} = x^{\frac{2}{2}} = x^1 = x$ then we get another answer. How is ...
1
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2answers
69 views

Writing $3.8473221018630726$ in the form $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$.

I attempted the following question on Brilliant which has to do with finding roots of a cubic polynomial. I was successful in finding what the only real root is but I am facing a problem rewriting the ...
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0answers
34 views

A modular equation of 29th degree of Dedekind’s $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(29i)}{\eta(i)}$ that is missing. Can someone help ...
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0answers
42 views

A modular equation of 23rd degree of Dedekind’s $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(23i)}{\eta(i)}$ that is missing. Can someone help ...
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0answers
34 views

A modular equation of 19th degree of Dedekind’s $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(19i)}{\eta(i)}$ that is missing. Can someone help ...
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vote
1answer
54 views

A modular equation of 17th degree of Dedekind’s $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(17i)}{\eta(i)}$ that is missing. Can someone help ...
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votes
1answer
64 views

A modular equation of 13th degree of Dedekind’s $ \eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(13i)} {\eta(i)}$ that is missing. Can someone help ...
2
votes
1answer
74 views

A modular equation of 11th degree of Dedekind's $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(11i)} {\eta(i)}$ that is missing. Can someone help ...
5
votes
5answers
784 views

Why isn't the definition of absolute value applied when squaring a radical containing a variable?

I recently learned about the following definition of absolute value: $|a| = \sqrt{a^2}$ Then I came across a solution to a problem that had the following step: $5 \geq \sqrt{5 - x}$ In order ...
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2answers
70 views

Solve $\left(\sqrt{\sqrt{2}-4-x}\right)+x^{\frac{1}{4}}=2^{\frac{-1}{4}}$

Solve the Equation in real/Complex numbers: Solve $$\left(\sqrt{\sqrt{2}-4-x}\right)+x^{\frac{1}{4}}=2^{\frac{-1}{4}}$$ My try: Letting $x=t^4$ we get We get $$\left(\sqrt{\sqrt{2}-4-t^4}\right)+...
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2answers
60 views

Find the minimum value without using derivative

Find the minimum value of $$f(x) = {3\over \sqrt{x}+1} - {12\over \sqrt{x}+3}$$ The domain of $f(x)$ is $x ∈ (0,∞)$. Then, using derivatives, I can find the minimum value is $f(1)=-1.5$. However, ...
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3answers
62 views

Find the minimum value of $x$ in the given that $\frac{\sqrt{2x^2-1} + \sqrt{x^2-1}}{\sqrt2x^2}=1$

How to simplify the given equation and find the minimum value of $x$ ? $$\frac{\sqrt{2x^2-1} + \sqrt{x^2-1}}{\sqrt2x^2}=1$$ I do square both sides but I doesn't make any sense
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1answer
36 views

non linear system of equations leading to quartic

The question asks to solve the system $$\begin{cases} \sqrt x + y = 16 \\ \sqrt y + x = 25 \end{cases}$$ Substitution leads to a fourth degree polynomial. Is there n easier way to solve it ? ...
3
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1answer
40 views

If $[K:\Bbb{Q}]=2$ then $K=\Bbb{Q}(\sqrt{d})$.

I am stuck on one question and sincerely have no idea how to proceed. Let $K$ be a field containing $\Bbb{Q}$ such that $[K : \Bbb{Q} ] = 2$. Prove that there exists a square free integer $d$ such ...
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0answers
37 views

Elementary Proof on Perfect Squares.

The Proof I'm working on is: If $r \in \mathbb{N}$ is not a perfect square, then $\sqrt{r}$ is irrational.\ The farthest I've gotten is by proving by contradiction, assuming that $\sqrt{r}$ is ...
1
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1answer
50 views

$\sin(\alpha) = \frac{\sqrt{n}}{k}$, where $n$ and $k$ are integers and $\alpha$ is a rational multiple of $\pi$

It is well known that the solutions of the equation $$ \sin\left(\frac\pi x\right)= \frac{\sqrt3}{2} $$ are $$ x=\frac{3}{6n+2}, n\in\mathbb{Z} $$ and $$ x=\frac{3}{6n+1}, n\in\mathbb{Z}. $$ ...
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1answer
19 views

How to stop graph from appearing in extra quadrants

I was simplifying the equation, $\sqrt{(x^2 + x)}xy = 5$, to, $\sqrt{(x^4 y^2 + x^3 y^2)} = 5$, using $\sqrt{(x)} x = \sqrt{(x^3)}$ but the graph appeared in all quadrants not just two and four. Can ...
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3answers
44 views

How to find the power series of $\sqrt{1+x^4}$?

The complete question is to find the integral from $0$ to $1$ of $$\sqrt{1+x^4}$$ I am unsure of how to find the power series of this equation in order to do that. I haven't dealt with square root ...
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1answer
52 views

inequality about $I=\int_{2012}^{3012}\sqrt[3]{x}\,dx$

Consider $$ \begin{align} & L=\sqrt[3]{2012}+\sqrt[3]{2013}+\ldots +\sqrt[3]{3011} \\ & R=\sqrt[3]{2013}+\sqrt[3]{2014}+\ldots +\sqrt[3]{3012} \\ \end{align}\ $$ and $$ I=\int_{2012}...
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1answer
48 views

If I pick $-1 = \sqrt{1}$, then why $ \sqrt{zw}= \sqrt{z}\sqrt{w} $ for only $z, w \le 0$?

This Reddit comment expatiates why the third equality (colored in red) is the one that's wrong in $\color{limegreen}{1 = \sqrt{1}} = \sqrt{(-1)(-1)} \color{red}{=} \sqrt{-1} \sqrt{-1} = i² = -1$. ...
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0answers
26 views

Proving the uniqueness of x=sqrt(r)

Given any $r \in \mathbb{R}_{>0}$, the number $\sqrt{r}$ is unique in the sense that, if $x$ is a positive real number such that $x^2 = r$, then $x = \sqrt{r}$ I would appreciate any nudge in the ...
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1answer
15 views

$3 \times 8$ Array Problem: Finding the Longest Line Containing Exactly 3 Dots

In the $3 \times 8$ array, the dots are evenly spaced horizontally and vertically with each dot 1 cm from the nearest neighboring dots. In simplest radical form, what is the number of units in the ...
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5answers
67 views

Square root without a calculator algorithm [duplicate]

Out of curiosity I'm trying to find an effective algorithm to find the value of a square root of a number(a) without a calculator. I'm trying to find a solution without searching it up. What I have ...
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2answers
33 views

Is $x^{\frac{1}{2}}$ equal to $\sqrt{x}$ or $\pm\sqrt{x}$?

I have seen that when graphing $f\left(x\right)=x^{\frac{1}{2}}$ the graph only outputs positive and zero values (the range is greater or equal to 0), but according to what I know about algebra (...
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1answer
69 views

Fractional parts of square roots of primes

To avoid confusion with other uses of braces, let $F:\Bbb R\to[0,1)$ be the fractional part function (usually noted as $\{\cdot\}$), so $F(x)=x-\lfloor x\rfloor$. It is known that the set $S:=\{F(\...
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1answer
19 views

Branch of $m$th root of a holomorphic function

Let $f$ be a holomorphic function in the open subset $G$ of $C$. Let the point $z_0$ of $G$ be a zero of $f$ of order $m$. I want to prove that there is a branch of $f^{1/m}$ in some open disk ...
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1answer
23 views

How can you express radicals as multiplication/addition?

How can you express radicals as multiplication/addition? Most mathematical operations clearly reduce to multiplication/addition (same thing), but how do you do that for exponentials/radicals? Thank ...
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3answers
94 views

estimate value of $\sqrt[30]{0.05}$

Yesterday I got an exam in which there was a problem and its solution results in $$\sqrt[30]{0.05}$$ I didn't go further calculation. Still I can't. My lecturer said, even I'm still not sure if he ...
2
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1answer
35 views

Near integers in powers of binomials with radicals

This question comes out of a mathematics calendar problem that asked for the tenths digit of the expression $(17 + \sqrt{280})^{17}$. The calendar implied the digit should be 9, but after playing ...
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0answers
25 views

$\sqrt 2$ in $F_p$ [duplicate]

Is there a way to find out if $\sqrt 2$ exists in $F_p$ depending on p?
2
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2answers
79 views

How can I solve this absolute value equation?

This is the equation: $|\sqrt{x-1} - 2| + |\sqrt{x-1} - 3| = 1$ Any help would be appreciated. Thanks!
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1answer
63 views

Arc length of $x^3 \sqrt{9-x}$ on $[0,9]$

This is supposed to be part of a student's Calc 2 homework; however, this seems to be an extremely difficult integration, and I couldn't figure it out. Find the arc length of $x^3 \sqrt{9-x}$ on the ...
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3answers
24 views

Solving radical and polynomial expressions

$2x^3 +3x^2 +2x+1 = x(2x+3)(\sqrt{x^2 + \frac{1}{x}} )$ Only solution i could find is x = -1, the LHS can be expressed as $(x+1)(2x^2+x+1)$ and the LHS has a $\sqrt{\frac{x^3+1}{x}}$ which has a ...
1
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0answers
65 views

Find $\sqrt{a} + \sqrt{b}\space$ as an exact answer for $\{a,b\} \in \mathbb R^+$

(Sorry if my MathJax is strange, I just skimmed through the tutorial and tried to make it work) I want to find what is basically a sum formula for square roots, similar how it exists for $\log(a) + \...
4
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0answers
82 views

On two nested radicals and divisibility

The last days I was playing around with two nested radicals which, as I learned here, can be simplified: $$u(x) =\sqrt{x + \sqrt{x +\sqrt{x +\sqrt{x +...}}}} = \frac{1}{2}(1+\sqrt{1+4x})$$ $$l(x) = \...