For questions involving radical numbers or expressions (i.e. expressions which involve $\sqrt[n]{\text{something}}$).

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4
votes
6answers
77 views

find x in $\sqrt[3]{6+\sqrt x} + \sqrt[3]{6-\sqrt x} = \sqrt[3] {3}$

Which one satisfies the equation $\sqrt[3]{6+\sqrt x} + \sqrt[3]{6-\sqrt x} = \sqrt[3] {3}$ (A)$27$ (B)$32$ (C)$45$ (D)$52$ (E)$63$ let $a = 6+\sqrt x , b=6-\sqrt x$ cube each side ...
3
votes
3answers
99 views

Is the inequality $| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ true?

I'm having some trouble deciding whether this inequality is true or not... $| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ for $x, y \in \mathbb{R}.$
3
votes
2answers
91 views

Proposition: $\sqrt{x + \sqrt{x + \sqrt{x + …}}} = \frac{1 + \sqrt{1 + 4x}}{2}$.

Proposition: $$\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = \frac{1 + \sqrt{1 + 4x}}{2}$$ I believe that this is true, and, using Desmos Graphing Calculator, it seems to be true. I will add how I derived ...
2
votes
0answers
41 views

Minima and maxima of the 6th degree polynomial are not expressible in radicals.

Question: Prove that there exists a polynomial $P$ with $\deg P \geq 6$ such that the minima and maxima are not expressible in radicals. I have the following proof: the minima and maxima of a 6th ...
9
votes
6answers
310 views

Proof that $\sqrt6 - \sqrt2 - \sqrt3$ is irrational.

I want to prove that: $$\sqrt6 - \sqrt2 - \sqrt3$$ is irrational. I have tried using squares, the $p/q$ definition of rationality and the facts that 1)rational$\times$ irrational=irrational (unless ...
0
votes
1answer
37 views

Why is a fractional exponent a root?

You learn very early that $\sqrt2 = 2^{\frac12}$ and $\sqrt[3]{8} = 8^{\frac13}$ but why is this? Usually when I ask this I get the answer, "Because it is defined that way" but is there any logical ...
2
votes
8answers
205 views

Value of an expression with cube root radical [closed]

What is the value of the following expression? $$\sqrt[3]{\ 17\sqrt{5}+38} - \sqrt[3]{17\sqrt{5}-38}$$
1
vote
2answers
44 views

Cubic equation $X^3+pX+q$ not solvable by radicals if $D=-4p^3 - 27q^2 >0$

How can one prove that the real cubic equation $$P(X)=X^3+pX+q$$ is not solvable by real radicals when $$D=-4p^3 - 27q^2 >0?$$ Which means that there is no sequence of extension: $$\mathbb R=L_0 ...
6
votes
5answers
86 views

is there a general formula for cases like $\sqrt{2}$ = $\frac {2}{\sqrt{2}}$?

I just noticed that $\sqrt{2}$ is equal to $\frac {2}{\sqrt{2}}$: $\sqrt{2} = 1.414213562$ $\frac {2}{\sqrt{2}} = 1.414213562$ It is confirmed by a hand-calculator. I tried to proof this as ...
3
votes
3answers
59 views

Negative roots of a cubic equation

Under what conditions will the cubic equation $ax^3 + bx^2 + cx + d$ where $a,b,c,d \in \mathbb R$ yield roots which have negative real parts? (All roots must have negative real parts) Motivation: I ...
-1
votes
1answer
33 views

Substitution in Binomial theorem

By substituting $0.08$ for $x$ in $(1+x)^{1/2}$ and its expansion to find $\sqrt 3$, correct to four significant figure. The answer is $1.732$ given by the practice. I couldn't find the connection by ...
0
votes
1answer
25 views

Approximating radicals using Binomial theorem

Use a suitable binomial expansion to find square root of $1.01$ and correct it to five decimal places. I use the formula $$(1+ax)= 1+ax + \frac{a(a-1)}{2!} + \cdots$$ but do not know where to stop. ...
6
votes
5answers
259 views

Computing as many digits as possible of $\sqrt{2}$ with a pen and a paper in 5 minutes

You have to compute as many digits as possible of $\sqrt{2}$ with a pen and a paper (an eraser if you're lucky...) in 5 minutes. What will you do? What is your justification for doing it? The ...
-2
votes
3answers
90 views

Find the next 3 terms of the seqence $1,2,\sqrt{7}, \sqrt{10}, \sqrt{13}, 4, \cdots $ [closed]

How do you find the next 3 terms of the sequence $1,2,\sqrt{7}, \sqrt{10}, \sqrt{13}, 4, \cdots $? I have not been able to even determine the type of sequence (arithmetic, geometric, or harmonic).
0
votes
3answers
87 views

Definite integral with the squared cosine under the square root

I can't solve this $$\int_{0}^{5}{\sqrt{1+\left(\dfrac{\pi}{2}\cos(10 \pi x)\right)^2}dx}$$ My approach: If $10\pi x =u \to 10\pi dx=du$, so ...
4
votes
3answers
58 views

Square roots equations

I had to solve this problem: $$\sqrt{x} + \sqrt{x-36} = 2$$ So I rearranged the equation this way: $$\sqrt{x-36} = 2 - \sqrt{x}$$ Then I squared both sides to get: $$x-36 = 4 - 4\sqrt{x} + x$$ Then I ...
0
votes
0answers
42 views

Can This Expression Be Simplified? (Involves Square Roots)

I started with the expression $$ \frac{4mlt(1-\sqrt{1-\frac{v^2}{c^2}})c^2}{\sqrt{1-\frac{v^2}{c^2}}} $$ and have ended up at: $$ \frac{4mlt(c^2 - c \sqrt{c^2-v^2})}{\sqrt{1-\frac{v^2}{c^2}}} $$ ...
0
votes
0answers
84 views

A good equation system

Given $a,b,c$ positive numbers, solve the system $\sqrt{xy}+\sqrt{xz}-x=a$, $\sqrt{yz}+\sqrt{yx}-y=b$ and $\sqrt{zx}+\sqrt{zy}-z=c$, where $x,y,z\in \mathbb{R}$. This only a pretty question. I did ...
3
votes
0answers
18 views

For any field $k$ and $n>0$: $X^n-a\text{ reducible }\ \Leftrightarrow\ a=b^p\ \vee\ a=-4b^4$.

I'm stuck on the following exercise: Let $k$ a field and $n$ a positive integer. Prove that $X^n-a\in k[X]$ is reducible if and only if there exist a prime $p$ dividing $n$ and a $b\in k$ such ...
10
votes
3answers
222 views

Prove or disprove that $\sqrt{2+\frac{1}{n}}$ is irrational for $n \in \mathbb{Z}^+$

I have good reason to suspect that $\sqrt{2+\frac{1}{n}}$ is irrational for all $n \in \mathbb{Z}^+$ but a proof of this eludes me. I've tried proof by contradiction have had no success. I've also ...
-1
votes
1answer
50 views

Why is there root of -1? [duplicate]

Always have we heard that there isn't a root of a negative number. That's why we call the root of -1 an imaginary number. But why, why do we need it and how even did they discover this case? I would ...
4
votes
1answer
80 views

If $x=(9+4\sqrt{5})^{48}=[x]+f$ . Find $x(1-f)$.

If $x=(9+4\sqrt{5})^{48}=[x]+f$, where $[x]$ is defined as integral part of $x$ and $f$ is a fraction, then $x(1-f)$ equals . $\color{green}{a.)\ 1} \\ b.)\ \text{less than}\ 1 \\ c.)\ ...
-4
votes
3answers
82 views

The limit of iterated square root with multiplication under the root, $\sqrt{ a \sqrt{ a \sqrt{a \cdots}}}$ [duplicate]

$$ \sqrt{ a \sqrt{ a \sqrt{a \cdots}}}=\text{ ?} $$ options were given as $0$ $-a$ $a$ $1$ i did not know how to solve it or what it was related to. Could anyone please explain the concept ...
1
vote
4answers
117 views

Why is $\sqrt{x^2}= |x|$ rather than $\pm x$? [duplicate]

Shouldn't the square root of a number have both a negative and positive root? According to Barron's, $\displaystyle \sqrt{x^2} = |x|$. I don't understand how.
3
votes
0answers
39 views

Can this cyclic septic be solved using only one 7th root extraction?

I. Quintics For an example of a cyclic quintic, we have for $p=11$, $$x^5 +x^4 −4x^3 −3x^2 +3x+1 = 0\tag1$$ The five roots $x_k$ for $k=0,1,2,3,4$, in radicals, are, $$x_k = ...
4
votes
2answers
38 views

Roots addition.

$$\sqrt{\frac{a+x^2}{x}-2\sqrt{a}}+\sqrt{\frac{a+x^2}{x}+2\sqrt{a}}=Q $$ One is expected to find $Q$ respecting $a>0$, $x>\sqrt{a}$ . I'd like to have my solution checked; namely the ...
0
votes
1answer
34 views

Solve the system of equations with one symmetrical equation

Solve the system of equations: $\left\{\begin{array}{l}x^3-y^3+(3x^2+y-2)\sqrt{y+1}-(3y^2+x-2)\sqrt{x+1}=0\\x^2+y^2+xy-7x-6y+14=0\end{array}\right.$ I used wolframalpha.com and got the solution: ...
5
votes
0answers
46 views

Solve the system of equations with $x=y$

Solve the system of equations: $\left\{\begin{array}{l}\sqrt{x^2+(y-2)(x-y)}+\sqrt{xy}=2y\\\sqrt{xy+x+5}-\dfrac{6x-5}{4}=\dfrac{1}{4}\left(\sqrt{2y+1}-2\right)^2\end{array}\right.$ I used ...
1
vote
4answers
78 views

What is $\sqrt{(-1)^2}$ [duplicate]

This question is primarily terminology based. In that $\sqrt{}$ denotes the principal square root. Here are two reasoning $\sqrt{(-1)^2}=1$ since $\sqrt{(-1)^2}=\sqrt{1}$ which we know has a ...
1
vote
2answers
44 views

Another (in)dependence over the nonzero rationals question

About one hour ago I asked a question which at first sight looked non-trivial to me but it is really trivial. Shame on me, whether I want it or not. Now I have, solely for fun, another question which ...
0
votes
1answer
26 views

Linear (in)dependence of roots over the nonzero rational numbers

I was reading some question on this site and stream of thought led me to the creation of another question that could be trivial for someone but I am unable even to start solving it. I wanna share this ...
1
vote
5answers
90 views

Sum of real solutions on equation $\sqrt{\sin^2{x} + {1 \over 2}} + \sqrt{\cos^2{x} + {1 \over 2}} = 2$ in interval $[0,2\pi]$ is?

I know that solution is $4\pi$ but I do not know how do they get to this solution. I always get that $x \in R$ and that $-1 < \cos 2x < 1$ when converting it to double angle. EDIT : So ok, I ...
3
votes
8answers
123 views

Find the cubic equation of $x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$

Find the cubic equation which has a root $$x=\sqrt[3]{2-\sqrt{3}}+\sqrt[3]{2+\sqrt{3}}$$ My attempt is ...
0
votes
4answers
65 views

$\lim_{n \to \infty} \sqrt[n] {n}$ [duplicate]

$$\lim\limits_{n \to \infty} \sqrt[n] {n}$$ Plotting $\sqrt[n] {n}$, it's clear that the limit is $1$. But I don't know how to proceed with this problem.
38
votes
10answers
3k views

Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then the values ...
8
votes
5answers
421 views

How to solve this equation for $x$?$\left(\sqrt{2-\sqrt{3}}\right)^x + \left(\sqrt{2+\sqrt{3}}\right)^x = 2$

This is probably such a beginner question (and it's not homework). I've stumbled upon this: $$\left(\sqrt{2-\sqrt{3}}\right)^x + \left(\sqrt{2+\sqrt{3}}\right)^x = 2$$ How to solve this equation for ...
1
vote
1answer
67 views

Solve the inequation: $5+3\sqrt{1-x^2}\geq x+4\sqrt{1-x}+3\sqrt{1+x}$

Solve the inequation: $5+3\sqrt{1-x^2}\geq x+4\sqrt{1-x}+3\sqrt{1+x}$ I tried to substitute $x=\cos t$ but don't get any result. Who can help me?
2
votes
5answers
65 views

Evaluate $\lim_{x\to 0}\frac{\sqrt[m]{\cos\alpha x}-\sqrt[m]{\cos\beta x}}{\sin^2x},m\in \mathbb{N}$

Evaluate $$\lim_{x\to 0}\frac{\sqrt[m]{\cos\alpha x}-\sqrt[m]{\cos\beta x}}{\sin^2x},m\in \mathbb{N}$$ I used L'Hospital's rule, but that didn't work. Could Taylor series be used? I don't know ...
1
vote
6answers
139 views

Limits and cube roots

I'm rather stumped at the moment. I can graph the following equation but I'm having trouble solving it algebraically. $$ \lim_{x\to 1} \frac{\sqrt[3]{x}-1}{\sqrt{x}-1} $$ Where do I start?
0
votes
2answers
43 views

Removing a radical from an addend in an equation?

Given this simple equation: $$ a + b\sqrt{5} = 7\sqrt{5} - 7 $$ I can really easily say that $a = -7$ and $b = 7$, because: $ a + b\sqrt{5} = 7\sqrt{5} - 7 \\ b\sqrt{5} + a = 7\sqrt{5} - 7 ...
0
votes
2answers
44 views

Proof that irrational coprime square root sums and products are always irrational?

I probably phrased it very bad. This is what I mean: $$\sqrt{x} + \sqrt{y} \neq R$$ x and y being non-square coprime natural numbers. And: $$\sqrt{xy} \neq R$$ x, y, AND R being coprime. Let's try to ...
7
votes
7answers
179 views

Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$

I used $$(n!)^{\frac{1}{n}}=e^{\frac{1}{n}\ln(n!)}=e^{\lim\limits_{n\to\infty}\frac{1}{n}\ln(n!)}$$ Then using Stirling's approximation and L'Hospital's rule on ...
0
votes
1answer
89 views

Solve the equation: $x^3+7x^2+16x+5=(1-2x)\sqrt[3]{-3x^2-7x+5}$

Solve the equation: $x^3+7x^2+16x+5=(1-2x)\sqrt[3]{-3x^2-7x+5}$ I used wolframalpha.com and get the solution: $x\in\{-3;2\sqrt2-3\}$ When $x=-3$, $\sqrt[3]{-3x^2-7x+5}=-1$ When $x=2\sqrt2-3$, ...
4
votes
1answer
72 views

Can $\sin(\pi/25)$ be expressed in radicals, revisited

This was inspired by this post. Let, $$q = e^{2\pi\, i/m}$$ D. Speyer's answer can be generalized as, $$\sin\Big(\frac{\pi}{m^2}\Big) = \frac{i}{2}\Big(-q^{1/(2m)}+q^{-1/(2m)} \Big)\tag1$$ while ...
2
votes
1answer
42 views

How to simplify a diabolical expression involving radicals

A friend and I have been working on this problem for hours - how can the following expression be simplified analytically? It equals $\frac{1}{2},$ and we have tried the following to no avail: ...
0
votes
0answers
28 views

upper and lower bounds related to $\sqrt[3]{a^3 + b}$ with $b \ll a$

We can use Taylor series to approximate $a + \frac{b}{3a}\approx \sqrt[3]{a^3 + b}$ with $b \ll a$. However, what are precise upper and lower bounds for this quantity? $$ a + \frac{b}{3a^2} < ...
7
votes
2answers
151 views

Show that $\lim\limits_{n \to \infty} \frac{(n!)^{1/n}}{n}= \frac{1}{e}$ [duplicate]

Show that $$\lim_{n \to \infty} \left\{\frac{(n!)^{1/n}}{n}\right\} = \frac{1}{e}$$ What I did is to let $U_n = \dfrac{(n!)^{\frac{1}{n}}}{n}$ and $U_{n+1} = ...
5
votes
5answers
126 views

What is the principal cubic root of $-8$?

According to my book it should be a real number, and according to WolframAlpha it should be $1+1.73i$ What is the correct answer?
2
votes
3answers
103 views

Get rid of the square roots of the denominator: $\dfrac{1}{\sqrt{7}-2\sqrt{5}+\sqrt{3}}$

How to get rid of the square roots of the denominator: $\dfrac{1}{\sqrt{7}-2\sqrt{5}+\sqrt{3}}$? I squared the whole denominator, but that didn't help. Also I searched for a propriety or ...
5
votes
2answers
88 views

Solve this equation: $(x+2)(\sqrt{2x+3}-2\sqrt{x+1})+\sqrt{2x^2+5x+3}=1$

Solve this equation: $(x+2)(\sqrt{2x+3}-2\sqrt{x+1})+\sqrt{2x^2+5x+3}=1$ This is my try Let $t=\sqrt{2x+3}-2\sqrt{x+1}$ or $t^2=6x+7-4\sqrt{2x^2+5x+3}=1$ The equation is equivalent to: ...