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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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5answers
85 views

Evaluate the limit $\lim_{x\to + \infty}\left(\sqrt{(x+a)(x+b)} -x\right)$ for $a, b \in \Bbb R$ [on hold]

Determine the limit of the following or prove it doesn't exist: $$ \lim_{x\to + \infty}\left(\sqrt{(x+a)(x+b)} - x\right) \space \space \text{where}\ \space a,b \in \mathbb R$$ I know that the ...
0
votes
4answers
66 views

If $a+\sqrt{a^2+1}= b+\sqrt{b^2+1}$, then $a=b$ or not? [duplicate]

It might be a silly question but if $$a+\sqrt{a^2+1}= b+\sqrt{b^2+1},$$ then can I conclude that $a=b$? I thought about squaring both sides but I think it is wrong! Because radicals will not be ...
0
votes
0answers
24 views

Exponent of the non-abelian group $1+J(FG)$, for finite non-abelian $p$-group $G.$

Let $G$ be a finite non-abelian $p$-group for some prime $p$, and $FG$ be group ring, where $F$ is a finite field of characteristic $p.$ I like to use proposition given below of the book "The ...
1
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1answer
70 views

Find the maximize value of $\sqrt{\frac{a}{a+c}}+\sqrt{\frac{b}{b+c}}-\frac{9\sqrt{c^2+1}}{8c}$

Let $a;b;c>0$ such that $ab+bc+ca=1$. Find the maximize value of $$K=\sqrt{\frac{a}{a+c}}+\sqrt{\frac{b}{b+c}}-\frac{9\sqrt{c^2+1}}{8c}$$ I can see: $$a=b=\frac{1}{\sqrt{7}};c=\frac{3}{\sqrt{7}}\...
0
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0answers
12 views

Proving that the Jacobson radical of a (not necessarily unital) ring is contained in the intersection of all left modular ideals.

I am following Lams book "A first course in non-commutative rings". I am attempting to prove that the Jacobson radical of a ring is precisely the intersection of all maximal modular left ideals of ...
0
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2answers
26 views

How to show that $\sqrt{1+ni}=\sqrt[4]{n^2+1}\cos\left(\frac{1}{2}\tan^{-1}(n)\right)+i\sqrt[4]{n^2+1}\sin\left(\frac{1}{2}\tan^{-1}(n)\right)$

Wolfram Alpha gives me that $$\begin{align} \sqrt{1+i}&=\sqrt[4]{2}\cos\left(\frac{1}{2}\tan^{-1}(1)\right)+i\sqrt[4]{2}\sin\left(\frac{1}{2}\tan^{-1}(1)\right)\\ &=\sqrt[4]{2}e^{\frac{1}{2}i\...
2
votes
4answers
60 views

Comparing $\ln 1000$, $\sqrt[5]{1000}$, $3^{1000}$, and $1000^{15}$ without calculator

In my Pre-Calculus class we were given the following problem: Put the following four values in order from smallest to largest: $\ln 1000$, principal $5$th root of $1000$, $3^{1000}$, and $1000^{15}$...
0
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5answers
92 views

When did Pythagoras's formula for the hypotenuse change from $\sqrt{a^2 + b^2}$ to $\sqrt{a^2 + b^2 + 2ab}$?

I was in secondary school in Nigeria in the 60s during the transitioning from colony to independence to republic. At school we were given this formula that is now burnt into my synapses because our ...
0
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0answers
29 views

given $p,q,r \ge 3$ study the diophantine equation $x^py^q=z^r-1$ using the $abc$-conjecture

I want to show that given $p,q,r \ge 3$ the diophantine equation $x^py^q=z^r-1$ has only finitely many solutions with $x,y,z \in \mathbb{N} = 1 ,2, \dots$ assuming the $abc$-conjecture. The proof ...
0
votes
2answers
47 views

How to make the nth root of a product act the same as simple multiplication in regard to parity?

I don't have any experience working with radicals, but I'm working on a function that requires products of nth roots to be positive or negative, depending on the number of negative factors. I've ...
0
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1answer
17 views

Simplify a square root by factoring it for sum-product method

I ran into a mathematical problem which I guess I can simplify by product-sum factoring, I learned this approach while studying in Kumon, but I am afraid I have forgotten. This is the expression: $...
-1
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1answer
58 views

Can we say that $i = -\sqrt{-1}$ because $i^2 =- 1$? [duplicate]

It is defined as $i^2 = -1$ then we can say $i=\pm \sqrt{-1}$ Then is true that $i$ can be equal to $ -\sqrt{-1}$
0
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3answers
39 views

Approximation of square root of sum of two squared terms

I have the following equation $\sqrt{(x_a-x_n)^2+(y_a-y_n)^2}$. I want to get rid of square-root and find an approximation which contains only $x_a,x_n,y_a,y_n$ (there should not be any other non-...
2
votes
3answers
149 views

help with $\;(x+3)^{1/3} = (x-1)^{1/2}$

Taking the 6th power of both sides leads to $x^2 + 6x + 9 = x^3 - 3x^2 + 3x -1 \;\Rightarrow\; 0 = (x-5)(x^2 + x + 2).$ Therefore, the only possible solutions are $\;\{5, \frac{-1}{2} \pm i\frac{\...
0
votes
1answer
52 views

Solve without a calculator: If $x+\sqrt{x}=13$ then $x+\frac{13}{\sqrt{x}}=?$

$$x+\sqrt{x}=13$$ $$x+\frac{13}{\sqrt{x}}=?$$ I tried to square(and also triple in another attempt) both sides of both of the equations hoping that I would find some expression to plug in. It didn'...
1
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0answers
27 views

Babylonian Method Limit Question

The Babylonian Method for finding square roots is a method that takes a guess, say $x$, and averages $x$ and $\frac{a}{x}$, where $a$ is the number you want to find the square root of. It then uses ...
0
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4answers
50 views

Put the $\sqrt[n]{n}$ numbers in to an increasing order

I had a test today and there was an extra problem I couldn’t solve. Put the $\sqrt[2]{2}, \sqrt[3]{3}, \dots, \sqrt[100]{100}$ numbers in to an increasing order. I just have no idea. I can handle ...
0
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1answer
27 views

Confusion on last step of Hardy's proof on the square root of “a rational number with imperfect square component”'s inability to be a rational number

Here is a copy of Hardy's proof: For suppose, if possible, that $p^2/q^2 = m/n$, $p$ having no factor in common with $q$, and $m$ no factor in common with $n$. Then $np^2 = mq^2$. Every factor of $...
0
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2answers
20 views

Factor square root out of quotient

How do I get from the first expression to the second? The only reason the limits are included is because on WolframAlpha it mentioned that this was the case for large negative numbers of x: $\lim_{x\...
2
votes
7answers
83 views

Help calculating $\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$

I need some help calculating this limit: $$\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$$ I know it's equal to 1 but I have no idea how to get there. Can anyone give ...
0
votes
6answers
41 views

Where is the logical flaw in solving this equation?

I ran across this equation... $\sqrt {2x+6}+4=x+3$ Without thinking, I solved for x in the following way: $\sqrt {2x+6}+4=x+3$ Subtract 4 from both sides. $\sqrt {2x+6}=x-1$ Square each side. ...
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votes
1answer
37 views

Is $(-1)^{1/3} = -1$ , but $(-1)^{2/6} = 1$. Why aren't these the same? [duplicate]

So if you try to solve $(-1)^{1/3}$ you can do $(-1)^{1/3} = \sqrt[3]{-1} = -1$ (cubic root of $-1$) But what if I write $1/3$ as $2/6$? $$(-1)^{1/3} = (-1)^{2/6}$$ So $(-1)^{2/6} = \sqrt[6]{(-...
5
votes
2answers
72 views

If $x>\sqrt{xy}>y$, then $x>y>0$.

I am trying to prove the following: If $x>\sqrt{xy}>y$, then show that $x>y>0$. My argument is as follows: We only need to show $y>0$. Suppose $y<0$. Then, for $\sqrt{xy}$ to be ...
5
votes
1answer
71 views

Has my conjecture using Pell's Equation been discovered before

We have $$\sqrt{d} = \frac{x}{y} - \frac{1}{f_0\cdot y} - \frac{1}{f_0\cdot f_1\cdot y}- \ldots - \frac{1}{f_0\cdot f_1\cdot\ldots\cdot f_n\cdot y}-\ldots\,,$$ where $$f_0 = 2x\,,$$ $$f_{n+1} = (f_n)^...
1
vote
0answers
44 views

$x(x − N p^2)(x + N p^2)(x^2 + N^2p^4) + p $ cannot be solved by radicals

I wish to show if for sufficiently large integers N, and a given prime p, the polynomial $x(x − N p^2)(x + N p^2)(x^2 + N^2p^4) + p $ can be solved by radicals. I think the answer is no but I am not ...
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votes
2answers
86 views

Is this a valid way to prove that $\sqrt 2$ is irrational?

Assume $\sqrt 2$ is rational. $$\sqrt 2= \frac{P}{Q}$$ $$2 = \biggr(\frac{P}{Q}\biggr)^2$$ $$2 = \frac{P^2}{Q^2}$$ $P^2 = 2Q^2$ which is impossible because there doesn't exist a number whose ...
0
votes
2answers
36 views

Simple equation with square root $2x-6-4\sqrt x=0$

Good evening to everybody, I have a doubt about the following simple equation: $$2x-6-4\sqrt x=0$$ I know that x should equal to $9$. But how do I arrive to this result? In particular, my ...
0
votes
1answer
19 views

Finding the critical point of this function and not getting expected answer.

I'm trying to find the critical point for f(x)=2x^2-1000*sqrt(x). However, I do already know the answer, but I have no idea how it is the answer. I've tried to work it out multiple times, but I keep ...
1
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6answers
68 views

Evaluating $\lim_{x\to3}\frac{\sqrt[3]{x+5}-2}{\sqrt[4]{x-2}-1}$ without L'Hopital

I have the following limit question, where different indices of roots appear in the numerator and the denominator $$\lim_{x\to3}\frac{\sqrt[3]{x+5}-2}{\sqrt[4]{x-2}-1}.$$ As we not allowed to use L'...
2
votes
2answers
40 views

How to solve this radical expression

I've been trying to solve this expression for at least two hours now... And I always get stuck towards the end, I don't know what I'm missing. $\frac 1{xy} \times (\sqrt{xy} - \frac{xy}{x-\sqrt{xy}})\...
1
vote
2answers
45 views

How do I derive root approximation functions?

I once saw a function for generating successively more-precise square root approximations, $f(x) = \frac{1}{2} ({x + \frac{S}{x}})$ where S is the square for which we are trying to calculate $\sqrt S$....
2
votes
1answer
130 views

Prove $(1+\sqrt{2})^k= \sqrt{n}+\sqrt{n-1}$ [duplicate]

Prove that for every $k \in \mathbb{N}$ there exists $n\in \mathbb{N}$ such that $(1+\sqrt{2})^k= \sqrt{n}+\sqrt{n-1}$ I tried to prove it using induction, but I could not move to the next step after ...
4
votes
3answers
42 views

Irrational equation

Solve over the real numbers: $$(x^2+x+1)^{1/3}+(2x+1)^{1/2}=2$$ I know for the second radical to be defined $x≥-0,5$ and I've attempted various methods I've solved other such equations with but to no ...
2
votes
4answers
71 views

Convergence to $\sqrt{2}$

It is a very good way to approximate $\sqrt{2}$ using the following; Let $D_{k}$ and $N_{k}$ be the denominator and the numerator of the $k$th term, respectively. Let $D_1=2$ and $N_1=3$, and for $k\...
1
vote
1answer
64 views

Proof for Natural number Identities

I am now trying to find proof for the following, which are significant to establishing proof for the Prime number relation that was originally stated in the question I posted here: $$\Bigl \lfloor \...
0
votes
1answer
103 views

Olympiad Inequality with Condition

I would like to prove this : Let $x,y,z$ be positive real numbers such that $xyz=1$ then we have : $$\frac{\left(\frac{1}{x}+\frac{z}{x}+z\right)\left(1+\frac{1}{x}+z\right)}{3\left(\frac{z}{x}\...
1
vote
5answers
59 views

Minimizing $\sqrt{(x+3)^2 + 49} + \sqrt{(x-5)^2 + 64}$

What is the minimum value of $$\sqrt{(x+3)^2 + 49} + \sqrt{(x-5)^2 + 64}$$ I tried getting the first derivative, but I can't solve the equation when I put $y = 0$. Methods without using calculus ...
1
vote
3answers
106 views

Is this a valid proof that $\sqrt{2}$ is irrational?

According to this math.stackexchange thread A number is a perfect square if and only if its prime factorization contains only even exponents. The prime factorization of $2 = 2^1$. Therefore, the ...
1
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0answers
29 views

Calculate $\operatorname{Rad}(A/\operatorname{Rad}(A))$ in a Banach algebra

Let $A$ be a Banach algebra with identity $e_A$, I'd like to find $\operatorname{Rad}(A/\operatorname{Rad}(A)).$ whre we define $\operatorname{Rad}(A)=\{a\in A:e_A-ba \in \text{InvA},b\in A\}...
1
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3answers
42 views

Create a set of N numbers with no common rational factor

Question: So I want to create a set of real numbers $\{a\}_{N} = \{a_{1}, a_{2}, \ldots, a_{N}\}$ such that if there exists a common factor between all of the elements, it must be irrational. In ...
4
votes
3answers
63 views

Finding dependencies such that $0 > \frac{2b^2r^2}{z}-\left(2r ^2-2br\sqrt{1-\frac{b^2}{z^2}}\right)z$

I'm trying to solve an inequality with 3 variables. $$0 > \frac{2 b^2 r^2}{z} - \left(2 r ^2 - 2 b r \sqrt{1 - \frac{b^2}{z^2}}\right) z$$ Basically, I want to know under which dependencies the ...
4
votes
4answers
110 views

How to prove $\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\ldots}}}}}}=\frac {\sqrt {4a-3}-1}2$

So, I was watching this video by blackpenredpen where he mentions that $$\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\ldots}}}}}}=\frac {\sqrt {4a-3}-1}2$$ so I wanted to try and prove it ...
0
votes
2answers
28 views

Does taking an n-th root of a negative number lead to contradictions?

$(-3)^3 = -27$. $f(x) = x^3$ is a bijective function, $g(x) = \sqrt[3]x$ is defined for all values of $x$. But my maths teacher said, that $\sqrt[n]a = b \iff b^n = a, a\geq0$. She said that allowing ...
2
votes
3answers
150 views

Prove $\sum_{1}^{\infty} \frac{1}{\sqrt{n}( n + \sqrt{n})} \lt 2 $

Prove that $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}( n + \sqrt{n})} \lt 2$$ I have found that $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}( n + \sqrt{n})} < \pi / 2 $ with integrating from $1$ to ...
2
votes
1answer
43 views

To prove any real Galois extension of $\mathbb{Q}$ of odd degree must not be radical

I'm trying to prove this using the statement: Let $K$ be a field containing n distinct roots of unity. An extension of $K$ of degree $n$ is a radical extension generated by an $n$th root of an ...
5
votes
3answers
184 views

Solve $\sqrt{\frac{4-x}x}+\sqrt{\frac{x-4}{x+1}}=2-\sqrt{x^2-12}$

The equation is $$\sqrt{\frac{4-x}x}+\sqrt{\frac{x-4}{x+1}}=2-\sqrt{x^2-12}$$ I tried squaring both left side and right side then bringing them to same numerator but got lost from there ... any ...
2
votes
0answers
65 views

How to simplify nested radicals?

How to simplify this expression?$$\sqrt{\smash[b]{18+\sqrt{260}}}-\sqrt{\smash[b]{12+\sqrt{140}}}-\sqrt{\smash[b]{20-2\sqrt{91}}},$$ which equals $0$. But how do I prove it? My attempt \begin{align}\...
-1
votes
2answers
50 views

limit $ \lim_{x \to 0} \frac{\sqrt{x}}{\sqrt{x+1} + \sqrt{x}} $ [closed]

$\lim_{x \to 0} \frac{\sqrt{x}}{\sqrt{x+1} + \sqrt{x}}$
0
votes
2answers
61 views

Prove or disprove inequality with 3 variables

I was trying to solve the inequality $$a-\sqrt[3]{a^3-c\cdot a^2}<b-\sqrt[3]{b^3-c\cdot b^2}$$ where $a>b>0$ and $c>0$. I managed to pack the part inside the cube root: $$a-\sqrt[3]{a^2(a-...
0
votes
3answers
74 views

If $0<x<y$, then prove that $\sqrt{x} <\sqrt{y}$ and $x <\sqrt{xy} <y$

In terms of Inequalities, we have covered till AM-GM inequality and Cauchy-Schwarz Inequality in class. I was thinking about applying AM-GM inequality but not sure if its right.