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

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0
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0answers
44 views

Existence of a root of $1+x$ [on hold]

How can I do the problem 1.3.9 b from Qing Liu's book "Algebraic Geometry and Arithmetic Curves". The problem goes as follows: Let $n\geq 2$ be an integer. Let $D=\mathbb Z[1/n]$. Let $A$ be a ...
4
votes
4answers
131 views

If $\sqrt{28x}$ is an integer is $\sqrt{7x}$ always an integer?

If $\sqrt{28x}$ is an integer is $\sqrt{7x}$ an integer? I have a book that says no, but I cannot think of an example of the contrary... Not looking for a full proof here just wanting to see a ...
1
vote
4answers
115 views

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

Why is $\sqrt{x^2 } = |x|$ ? Squaring always produces a positive result, and you obviously square the equation before taking the root of it. So where exactly is the problem?
4
votes
4answers
80 views

Calculate simple expression: $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$

Tell me please, how calculate this expression: $$ \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} $$ The result should be a number. I try this: $$ \frac{\left(\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - ...
1
vote
6answers
132 views

Prove the existence of the square root of $2$.

I am trying to prove the existence of the square root of $2$. I have some steps with a very vague explanation and I would like to clarify. The proof: Let $$S=\{x\in\mathbb R\mid x\geqslant 0 \text{ ...
0
votes
3answers
53 views

Simplifying Cube Roots Containing a Square Root

I was doing a problem today, and arrived at the (correct) answer of $x^3 = 16000\sqrt2$ Obviously I want to simplify this further. My text book jumps straight to $x = 20\sqrt2$ with no explanation. ...
2
votes
3answers
93 views

Computing $\sqrt[3]{1\,}$

I know that the answer is always $1$, but they are looking for some way to get to that answer and I don't know what it is. I am not good at english math terms, but maybe it has to do with differential ...
5
votes
3answers
99 views

Which is more simplified: $a\sqrt{b}$ or $\sqrt{c}$?

Which is considered more simplified (if it matters)? $a \sqrt{b}$ or $\sqrt{c}$ For example: $2 \sqrt{3}$ or $\sqrt{12}$
3
votes
3answers
75 views

Prove that $\sqrt{3}$ is not a rational number [duplicate]

There is a similar question however that question asks why $3 |p^2$. Here the question is about $ 3 | p^2 \rightarrow 3 | p$. It is a simple exercise (1.2.1) from Abbot's "Understanding Analysis". ...
1
vote
3answers
29 views

Rationalization of a fraction

Can someone please explain how i can rationalize the fraction $\frac{2 +\sqrt{3}}{(2-\sqrt{3})^3}$ so that i obtain an answer that has an exponent of 6. I basically need to compare this value with ...
1
vote
3answers
23 views

What should the initial guess be for the Bablyonian method of calculating square roots?

You can use any value as the initial guess for the Babylonian method of calculating a square root (other than 0), but the closer the guess to the root, the more accurate your result per iteration. Of ...
1
vote
0answers
44 views

Evaluating Continued Product of Radicals

I am trying to find the following continued product $y = \sqrt{p}.\sqrt{p+p.\sqrt{p}}. \sqrt{p+p.\sqrt{p+p\sqrt{p}}}...........\infty$ for $p=\frac12$ First I squared the LHS and RHS to get $y^2 = ...
0
votes
1answer
39 views

Finding a positive lower bound of hte sequence $\frac{\sqrt[n]{n!}}n$

I am given a sequence {(n'th root of n!)/n}. Can I show that the sequence is bounded below by a real no. which is greater than 0, by not calculating the limit of it....???thank you
1
vote
0answers
26 views

Is it possible that some combination of partial sums of $\sum \sqrt{k}$ be a rational number? $\text{where } k \text { not a perfect square}$

Is it possible that some combination of partial sums of $\sum \sqrt{k}$ be a rational number? $k=2,3,5,\cdots \text{where } k \text { not a perfect square}$ More ever : can a linear combination of ...
0
votes
1answer
53 views

Square root of whole number number of solutions [duplicate]

Hi The GRE prep test is asking for the square root of a number.. for example $\sqrt{16}$. It says the answer is $4$. Couldn't the solution be both $4$ and $-4$?
0
votes
1answer
50 views

Solve limit using Stolz's theorem: $\lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}$ [closed]

Solve this limit usinig Stolz's theorem. Any help?! $$\lim\limits_{n\rightarrow \infty} \frac{n}{\sqrt[n]{n!}}$$
7
votes
2answers
164 views

Value of $\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}}$

Here is the question: $$\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}} = \;?$$ (original image) I ...
2
votes
3answers
60 views

Proof of the irrationality of $\sqrt n$, where $n$ is square free

I am trying to review some old algebra, and in particular I wanna show that $\sqrt2$ is irrational Since integers are the only integral elements of $\mathbb Q$ over $\mathbb Z$, assume $r=\sqrt 2$ is ...
3
votes
2answers
71 views

Simplifying $\frac{1}{\sqrt{-1}}$ [duplicate]

When trying to simplify $\frac{1}{\sqrt{-1}}$, you could rationalize it: $$\frac{1}{\sqrt{-1}}\cdot\frac{\sqrt{-1}}{\sqrt{-1}}=\frac{\sqrt{-1}}{-1}=-\sqrt{-1}$$ Or you could simplify it as one ...
1
vote
3answers
38 views

Show that $1/\sqrt{1} + 1/\sqrt{2} + … + 1/\sqrt{n} \leq 2\sqrt{n}-1$ [duplicate]

Show that $1/\sqrt{1} + 1/\sqrt{2} + ... + 1/\sqrt{n} \leq 2\sqrt{n}-1$ for $n\geq 1$ I attempted the problem but I get stuck trying to show that if the statment is true for some $k\geq1$ then $k+1$ ...
1
vote
2answers
342 views

algebra problem, Solve the equation [closed]

a nice problem: Solve the equation $$\left|2x-57-2\sqrt{x-55}+\frac{1}{x-54-2\sqrt{x-55}}\right|=|x-1|.$$ It's just for sharing a new ideas, thanks:)
2
votes
5answers
73 views

What will change if we admit a different definition of $\sqrt a$

We know that $\sqrt a$ is the non negative solution of the equation $x^2=a$ with $a\geq 0$. So if we want to solve the equation $x^2=a$, we say that $x=\pm\sqrt a$. How will mathematics be affected ...
9
votes
1answer
132 views

Why does the elliptic curve for $a+b+c = abc = 6$ involve a solvable nonic?

The curve discussed in this OP's post, $$\color{brown}{-24a+36a^2-12a^3+a^4}=z^2\tag1$$ is birationally equivalent to an elliptic curve. Following E. Delanoy's post, let $G$ be the set of rational ...
9
votes
6answers
165 views

Follow-up Question: Proof of Irrationality of $\sqrt{3}$

As a follow-up to this question, I noticed that the proof used the fact that $p$ and $q$ were "even". Clearly, when replacing factors of $2$ with factors of $3$ everything does not simply come down to ...
1
vote
3answers
101 views

Prove the inequality - inequality involving surds

Prove that for $r$ greater than or equal to 1: $\displaystyle 2(\sqrt{r+1} - \sqrt{r}) < \frac{1}{\sqrt{r}} < 2(\sqrt{r}-\sqrt{r-1})$ Any help on this would be much appreciated.
0
votes
1answer
47 views

Solving algebraic equations with radicals

I have several problems requiring assistance. Solve for x: $x\left( x-\sqrt { 3 } \right) \left( x+1 \right) +3-\sqrt { 3 } \quad =\quad 0$ I've followed the suggestion to get x^2 - (√3 -1)x + ...
0
votes
3answers
68 views

Solve equation involving square roots [closed]

Please help me solve this equation for $x$: $$\sqrt{\sqrt{x+4} + 4} = x$$
1
vote
2answers
46 views

Take $\sqrt[n]{m}$ where $n$ is a positive non-integer number. Is this possible?

The title pretty much says it all. I am trying to figure out how to take the $n$-th root of a number $m$ where $n$ is a fractional number. Unfortunately all I could turn up on Google was how to take a ...
0
votes
3answers
35 views

Radical under Radical expression

how to find the sum of $\sqrt{\frac54 + \sqrt{\frac32}} + \sqrt{\frac54 - \sqrt{\frac32}} $ ? Is there a method to solve these kind of equations ?
2
votes
2answers
45 views

Proving $\sum_{j=1}^n \frac{1}{\sqrt{j}} > \sqrt{n}$ with induction

Problem: Prove with induction that \begin{align*} \sum_{j=1}^n \frac{1}{\sqrt{j}} > \sqrt{n} \end{align*} for every natural number $n \geq 2$. Attempt at proof: Basic step: For $n = 2$ we have $1 ...
8
votes
3answers
186 views

Why does $x^2+47y^2 = z^5$ involve solvable quintics?

This is related to the post on $x^2+ny^2=z^k$. In response to my answer on, $$x^2+47y^2 = z^3\tag1$$ where $z$ is not of form $p^2+nq^2$, Will Jagy provided one for, $$x^2+47y^2 = z^5\tag2$$ as, ...
1
vote
1answer
71 views

Are there expansions of the expression $(a+b)^{1/n}$? [duplicate]

Is there an expansion of the expression in the bracket such as $$ \sqrt{a + b} = (a + b)^{1/2}$$ If not do you know of a method that lets us solve such expression and ones with higher roots?
6
votes
5answers
134 views

$p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.

I want to prove that for $p,q,r$ different primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational. Is the following proof correct? If $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is rational, then ...
1
vote
1answer
51 views

Two infinite radicals question

Hello I have stucked with theese two questions: $\sqrt{a:\sqrt{a:\sqrt{a: \cdots}}} + \sqrt[3]{a\cdot\sqrt[3]{a\cdot\sqrt[3]{a\cdots}}} = 12$ $a=\text{ ?}$ ...
1
vote
4answers
142 views

A basic root numbers question

If $\sqrt{x^2+5} - \sqrt{x^2-3} = 2$, then what is $\sqrt{x^2+5} + \sqrt{x^2-3}$?
6
votes
2answers
131 views

The value of the integral $\int_0^2\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$

The value of definite integral $$\int\limits_{0}^{2}\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$$ is $$(A)\,4 \quad(B)\,5 \quad (C)\,6 \quad(D)\,7$$ My attempt: I tried using ...
2
votes
3answers
86 views

Solve the equation $4\sqrt{2-x^2}=-x^3-x^2+3x+3$

Solve the equation in $\Bbb R$: $$4\sqrt{2-x^2}=-x^3-x^2+3x+3$$ Is there a unique solution $x=1$? I have trouble when I try to prove it. I really appreciate if some one can help me. Thanks!
4
votes
6answers
90 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
102 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
115 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 ...
7
votes
3answers
224 views

Abel-Ruffini theorem, Galois theory and minima and maxima

Questions: Does there exist a proof of the Abel-Ruffini theorem without using Galois theory? Does there exist a proof that there exists a polynomial $P$ with $\deg P = 5$ such that the roots are not ...
2
votes
3answers
92 views

Explain the proof that the root of a prime number is an irrational number

Though the proof of this is done in a previous question i have a doubt in a certain concept so i ask to clear it.In the proof we say that $\sqrt{p} = \frac{a}{b}$. (In their lowest form.) $p = a^2 ...
10
votes
6answers
366 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
39 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
218 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
46 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
64 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
36 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
28 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. ...