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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
33 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
49 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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2answers
30 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
45 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
46 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
13 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
62 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
32 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
68 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
91 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 ...
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1answer
22 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
23 views

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

Is there a way to find out if $\sqrt 2$ exists in $F_p$ depending on p?
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2answers
78 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
62 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 ...
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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) + \...
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0answers
65 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) = \...
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1answer
60 views

How to solve decics like $x^{10}+100x^2+160x+64=0$ having Galois group 10T33?

Using the approach described in Smart way to solve octics like $x^8+5992704x-304129728=0$ (the method DecomPoly available in GAP) the decic quadrinomial from this ...
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1answer
18 views

Order theory between radicals [closed]

Consider $p$ and $q$ $∈$ $N$ where $p>q$. What is the order between the numbers $\sqrt{2pq}$ and $\sqrt{p}+\sqrt{q}$?
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4answers
878 views

Is $\sqrt{\sin x}$ periodic?

$\sin^2(x)$ has period $\pi$ but it seems to me $\sqrt{\sin x}$ is not periodic since inside square root has to be positive and when it is negative, it is not defined. Does it creates problem ...
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1answer
37 views

How to find the next number in a sequence whose square root is a whole number

I've just rephrased the question a bit. S(i=1 to n) is a set of n whole numbers where Square_Root(Si) is always a whole number. ...
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3answers
54 views

Approximating series of fractions [duplicate]

Let $$ P = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}+ \frac{1}{\sqrt{4}} ... +\frac{1}{\sqrt{10000}}$$ what is the value of the floor function of P? My try: I tried assuming these 2 bounds $$ P_x =...
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1answer
42 views

Square root and Cube root in the same equation [closed]

In a recent test I took, I received a question in an awkward form: $$\sqrt y-\sqrt[3]{1000-y}=16$$ How would I go about solving this?
2
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2answers
49 views

Evaluating expression with Integer part and Fraction part of a nested radical

Let $$A= \sqrt{93+28\sqrt{11}}$$ if $B$ is the integer part of $A$ and $C$ is the fraction part of $C$, what is the value of $$B+C^2$$ I tried manipulating it by setting $$ A=B+C$$ but I can'...
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1answer
68 views

Convergence of $\sqrt[n]{(1+x)^n-(1+x)+\sqrt[n]{(1+x)^n-(1+x)+\sqrt[n]{\cdot\cdot\cdot}}}$

If one writes $$1+x=\sqrt{(1+x)^2}=\sqrt{1+2x+x^2}=\sqrt{x+x^2+(1+x)}$$ then one has a recursive definition of the function $1+x$ which can be used to write $1+x$ as the infinite nested radical: $$1+x=...
2
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1answer
51 views

If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals.

If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals. My attemt: Any general equation of degree $\geq 5$ is not solvable by radicals. ...
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0answers
29 views

How can I prove that $n \sqrt{\frac{x}{n^2}} = \sqrt{x} | n \in \mathbb{N}$?

I came across this observation in an exam today, and thought that this might be useful in making certain algorithms run faster, but first I want a way to prove that this is true. How can I do this? ...
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1answer
42 views

Solving algebra with multiple square root

I am currently solving an algebra and can't figure it out, could anyone help me on this? $$2\sqrt{N + \sqrt{N^2+4c^2}} = \sqrt{N + \sqrt{N^2+3c^2}} + \sqrt{N + \sqrt{N^2+5c^2}}$$ Which I would like ...
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1answer
24 views

How to break long expressions into multiple lines?

Say I want to write a long radical expression like this $\sqrt{a + b - c + d - e + f - g + h - i + j…}$ on paper or blackboard. It cannot fit in one line. How should I break it into two lines? ...
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2answers
45 views

Which simplified form of $-2\sqrt[3]{-250}$ is correct?

I simplified the cube root expression into two: $10\sqrt[3]{2}$ and $-10\sqrt[3]{-2}$. Both yield the same approximation when I solved them using calculator. Which is correct? Or both should be ...
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3answers
38 views

Optimization without complex numbers

We need to find a minimum of functions: (1+$\sqrt x$)$^2$+$y^2$ Due to the fact that the function has a square root, the optimization algorithm goes into the area of complex numbers. How to make so ...
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4answers
74 views

Solve the following equation: $\sqrt{4x^2 - 15x + 20} = 4x - 10 + 7\sqrt{x - 1}$.

Solve the following equation. $$\large \sqrt{4x^2 - 15x + 20} = 4x - 10 + 7\sqrt{x - 1}$$ I try to let $2x - 6 = a$ and $\sqrt{x - 1} = b$. Then the equation becomes: $$\sqrt{a^2 + 9b^2 - 7} = 2a + ...
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3answers
53 views

Multiplication of the square root of complex numbers

$$ \sqrt{-21+20i}.\sqrt{-21-20i}=\pm(2+5i).\pm(2-5i)=\pm29 $$ But why it is not $$ \sqrt{-21+20i}.\sqrt{-21-20i}=\sqrt{(-21+20i)(-21-20i)}=\sqrt{|z|^2}\\ =\sqrt{441+400}=\sqrt{841}=29 $$ Does this ...
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3answers
66 views

How to simplify $\sqrt{2+\sqrt{3}}$ $?$

Simplify $\left(\frac{2(\sqrt2 + \sqrt6)}{3(\sqrt{2+\sqrt3}}\right)$ The answer to this question is $\frac{4}{3}$ in a workbook. How would I simplify $\sqrt{2+\sqrt3}$ $?$ If it was something like $...
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6answers
85 views

Solve the equation $x^2 + 4(\sqrt{1 - x} + \sqrt{1 + x}) - 8 = 0$

Solve the equation $x^2 + 4(\sqrt{1 + x} + \sqrt{1 - x}) - 8 = 0$. Let $\sqrt{1 + x} = a$, $\sqrt{1 - x} = b$. I tried doing this. "$1 - x^2 = [\sqrt{(1 - x)(1 + x)}]^2 = (ab)^2$. The original ...
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3answers
77 views

What went wrong in proving $i=1$ [duplicate]

I started with $$x=(-16)^{\frac{1}{2}}$$ $$x=(-16)^{\frac{2}{4}}$$ Since $$(a^m)^n=a^{mn}$$ we have: $$x=((-16)^2)^{\frac{1}{4}}$$ $$x=((16^2)^{\frac{1}{4}}$$ $$x=\sqrt{16}=4$$ Hence $$(-16)^{\...
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1answer
44 views

Simplification of identity with square roots [closed]

$\dfrac{\sqrt{x + 1}}{2x + 1} + \dfrac{\sqrt{2x + 1}}{x + 1} = 1 \tag 1$ How can I find the value of $x$ in this question?
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1answer
32 views

Range of radical functions.

Suppose we have $f(x)=\sqrt{x-1}+\sqrt{5-x}$; how do we find range for this function? Single radicals are easy , but two of them are in this particular function.I have the domain of the function and ...
2
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3answers
86 views

Solve a system equation in $\mathbb{R}$ - $\sqrt{x+y}+\sqrt{x+3}=\frac{1}{x}\left(y-3\right)$

how to solve a system equation with radical $$\sqrt{x+y}+\sqrt{x+3}=\frac{1}{x}\left(y-3\right)$$ And $$\sqrt{x+y}+\sqrt{x}=x+3$$ This system has $1$ root is $x=1;y=8$,but i have no idea which is ...
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2answers
41 views

How does this algorithm calculates the natural logarithm?

This is the pseudocode ...
2
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1answer
38 views

Efficient method to calculate a big n-th root of a number

I need to implement an algorithm to calculate big n-th roots like $\sqrt[\leftroot{-2}\uproot{2}\mbox{13500}]{ 200}$. I have tried the Newton's method but it requires the calculation of very big ...
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0answers
44 views

Infinitely Nested Radical with Fibonacci Coefficients

I was wondering if the following infinitely nested radical can be evaluated. $x= \sqrt{1+ \textbf{1}\sqrt{1+ \textbf{1}\sqrt{1+ \textbf{2}\sqrt{1+ \textbf{3}\sqrt{1+ \textbf{5}\sqrt{1+ \dots }}}}}} $ ...
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1answer
47 views

Simplifying a expression which involves a square root: $\sqrt{36 - 4x^2}$

I know that $ 2 \sqrt{9-x^2}$ is the alternate form for $\sqrt{36 - 4x^2}$. I tried but i didn't figure out how to get there. Can someone help?
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2answers
25 views

Reference on polynomial equation that demonstrates the history of complex numbers

Pardon my ignorance and lack of thorough understanding, but I am missing a piece of the puzzle when it comes to complex numbers and can't seem to find an answer. I have been trying to understand ...
2
votes
3answers
67 views

$6\sqrt{24} + 7\sqrt{54} - 12\sqrt{6}$ = $21\sqrt{6}$ but I get $207\sqrt{6}$

I'm asked to simplify $6\sqrt{24} + 7\sqrt{54} - 12\sqrt{6}$ The provided solution is $21\sqrt{6}$ but I arrive at a different amount. Here is my working, trying to understand where I went wrong: ...
1
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1answer
34 views

Why is the root of a square restrictive in my calculation?

I have a parabola $f(x) = ax^2 + bx + c$, where $a<0$, and I want to find the domain where $f$ is greater than a secant defined by taking a point $w$ on the domain, and drawing the secant from $(w-...