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

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1
vote
4answers
53 views

Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$

How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$. Please help me. Thanks!
3
votes
5answers
96 views

When $a\to \infty$, $\sqrt{a^2+4}$ behaves as $a+\frac{2}{a}$?

What does it mean that $\,\,f(a)=\sqrt{a^2+4}\,\,$ behaves as $\,a+\dfrac{2}{a},\,$ as $a\to \infty$? How can this be justified? Thanks.
1
vote
2answers
46 views

Solving the system $a^2-6=2\sqrt{2c+6}, \, b^2-6=2\sqrt{2a+6}, \, c^2-6=2\sqrt{2b+6}$

Question: Solve the following system for $a,b,c\in \mathbb{R}$: $$\begin{cases} b^2-6=2\sqrt{2a+6}\\ c^2-6=2\sqrt{2b+6}\\ a^2-6=2\sqrt{2c+6} \end{cases}$$ I found the following:$$ ...
1
vote
1answer
25 views

Branches of the square root function in the domain $D=\mathbb{C}$\ $[0,\infty)$

I saw the solution for this in Palka's book and one of the branches was defined as follows. $$ g(z) = \begin{cases} \sqrt{z}, & z\in D ,Im(z)\geq0 \\ -\sqrt{z}, & z\in D ,Im(z)<0 ...
0
votes
2answers
40 views

infinite limit question from Calc I

Find the limit $$\lim_{x\to\infty}\sqrt{x^2+x+1}-x$$ This limit is part of a question involving squeeze theorum, the limit is $\frac12$ but i don't know how to prove it because of the polynomial in ...
3
votes
3answers
31 views

$\lim_{x \rightarrow \infty} \sqrt{(8x^2-3)/(2x^2+x)} $

$$ \lim_{x \rightarrow \infty} \sqrt{\frac{8x^2-3}{2x^2+x}} $$ Do I square the whole function? I don't know how to start.
1
vote
1answer
25 views

Simplifying square roots?

How would I simplify $\sqrt{\frac{800}{3}}$ preferably by a factor tree? I know it simplifies into $\frac{20\sqrt{6}}{3}$. I just don't know the steps to get there. Help please?
-1
votes
2answers
30 views

Make $l$ the subject of $s=2π\sqrt l/32$ [closed]

The formula $s=2π\sqrt l/32$ represents the swing of a pendulum, where s is the time, in seconds, to swing back and forth, and l is the length of the pendulum, in feet. Solve for l. Thanks to ...
0
votes
3answers
48 views

How to simplify $\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$

I in trouble simplifying this: $$\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$$ couldn't find a solution. Can you help?
0
votes
1answer
37 views

Need help with simplifying a radical expression

I need help with simplifying this radical expression: $\sqrt{(5+2\sqrt{6})}(49-20\sqrt{6})(9\sqrt{3}+11\sqrt{2})$.
1
vote
0answers
71 views

Any general hints on how to prove that two functions$\ f(n)$ and$\ g(m_1,m_2,…,m_{28})$ never have a common natural divisor?

All the variables are natural numbers. I'm not asking for a proof, since while we simply have$\ f(n)=n^3-n+1$,$\ g$ is a very long sum of cube roots (which contain square roots as well). I'm after ...
0
votes
2answers
85 views

Is $\sum i^{1/i}$ bounded?

I'm trying to find the limit $$ \lim_{n\to\infty}\sum_{i=1}^n \frac{i^{1/i}}n\,. $$ I was going to say that $\lim_{n\to\infty} \frac1n=0$ and $\sum i^{1/i}$ is bounded but I can't prove it.
2
votes
3answers
92 views

Prove that $2\sqrt{n}\sqrt{n+1} < 2n + 1$ for all positive integers.

I've been testing this with many values and it seems to always be true. I've been trying to rework the inequality into a form where it's much more obvious that the left hand side is always less than ...
0
votes
1answer
34 views

How to show that$\ \sqrt[3]{ \sqrt{y^2-x}+y}+ \sqrt[3]{-\sqrt{y^2-x}+y} = k \implies y = \frac{k\left(k^2-3 \sqrt[3]{x}\right)}{2}$?

We also have $\ x \ne y $, $\ y > 1$, $\ 0<x<1$,$\ k \ne 0$. I have tried on my own, by canceling out the roots, but they keep on appearing. I guess that is not the right way. Thanks in ...
3
votes
2answers
48 views

Simplifying a square root fraction

Simplify the following $$\frac{\sqrt{3}}{\sqrt{2}(\sqrt{6} - \sqrt{3})}$$ Apparently the answer is $\frac{1}{2} (2 + \sqrt{2})$ but can't for the life of me see how to get it. Any help is massively ...
0
votes
1answer
71 views

How can I find x and z if: $\sqrt{(x-20)^{2} + (5-30)^{2} + (z-40)^{2}} = 100$ and $x \sqrt\frac{1}{6} + 5\sqrt\frac{1}{3} + z \sqrt\frac12= 0$?

How can I find x and z if: $\sqrt{((x-20)^{2} + (5-30)^{2} + (z-40)^{2})} = 100$ and $\left(x\times \sqrt\frac{1}{6} + 5\times \sqrt\frac{1}{3} + z\times \sqrt\frac{1}{2}\right) = 0$ ?
2
votes
4answers
265 views

How to deduce the following trig relation?

How can I deduce: $$\sqrt{|x|}\sin(\frac{1}{x}) \le \sqrt{|x|}$$?? I know of the relation. $$\sin(u) \le u$$ $$u = \frac{1}{x}$$ $$\sin(1/x) \le \frac{1}{x}$$ But nothing related to $\sqrt{x}$ ...
5
votes
4answers
64 views

Find the limit as x approaches negative infinity for $\sqrt{x^2+x-1} +x$

Find the limit as x approaches negative infinity for $\sqrt{x^2+x-1} +x$ My solution: multiplying by: $\displaystyle\frac{\sqrt{x^2+x-1}-x}{\sqrt{x^2+x-1}-x}$ Which gives us: ...
3
votes
0answers
31 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
6
votes
7answers
230 views

$ f^{7} $ is holomorphic implies that $ f $ is holomorphic. [closed]

I need some help with this problem: Let $ \Omega $ be a domain. If $ f: \Omega \to \mathbb{C} $ is a continuous function and $ f^{7} $ is holomorphic on $ \Omega $, then $ f $ is also holomorphic ...
3
votes
1answer
59 views

Why doesn't this method of solution work?

Solve $$\sqrt{2x^2 - 7x + 1} - \sqrt{2x^2 - 9x + 4} = 1 \tag1$$ I tried to do the following: $$(2x^2 - 7x + 1) - (2x^2 - 9x + 4) = 2x-3\tag2$$ Dividing $(2)$ by $(1)$ yields $$\sqrt{2x^2 ...
1
vote
0answers
64 views

Nestedradicals found in the solution of equation $x^{257}=1$

I was looking for the exact solutions of $\cos\frac{2\pi}{257}$, it lead me to the following expressions. ...
18
votes
5answers
409 views

Find the value of $\sqrt{10\sqrt{10\sqrt{10…}}}$

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it ...
0
votes
3answers
38 views

How can I evaluate this limit?

I'm studying for an upcoming midterm and i'm stuck on this question. It's asking me to evaluate the following limit and justify my answer. $\lim \limits_{x \to \infty} \sqrt{x^2+3x} - \sqrt{x^2-2x}$ ...
2
votes
1answer
50 views

How prove this$\lfloor \sqrt{2x-\lfloor\sqrt{2x}\rfloor}\rfloor=\lfloor\frac{\sqrt{8x+1}-1}{2}\rfloor$

Question: let $x\ge 0$, show that $$\lfloor \sqrt{2x-\lfloor\sqrt{2x}\rfloor}\rfloor=\lfloor\dfrac{\sqrt{8x+1}-1}{2}\rfloor$$ My idea: let $\lfloor \sqrt{2x}\rfloor =m$ then ...
1
vote
2answers
28 views

Relationship between Difference of Two Numbers and Their Square Roots

Is there a relationship between the difference of two numbers and the difference of their square roots? For example, can we say that ${| \sqrt x - \sqrt y|\leq |x - y|}$ when ${ x, y \geq 1 }$, but ...
9
votes
4answers
179 views

Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$

How could we prove that there does not exist a positive integer $n$, so that the number $$({\sqrt{2}+1})^{1/n} + ({\sqrt{2}-1})^{1/n}$$ is rational? I think it could be done inductively from a more ...
0
votes
3answers
39 views

The limit of $\sqrt{x^2+x+1}-\sqrt{x^2+1}$ as $x\to\infty$ [closed]

Currently I'm self studying limits. but I don't know how to get the answer to this question: $$\lim _ { x\to \infty }\left(\sqrt{x^2+x+1}-\sqrt{x^2+1}\right)$$ can someone help me
4
votes
1answer
59 views

Algebra problem solve for a,b,c and d?

Can anyone find the values of these integers: a,b,c and d? $$1+\sqrt{2}+\sqrt{3}+\sqrt{6} = \sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d}}}}$$ a+b+c+d = ? Thank you.
1
vote
2answers
43 views

Holomorphic branch of the $n^{\mathrm{th}}$ root of $f$

The problem states to prove that if $h$ is a branch of $f^{1/n}$ for integer $n > 0$ (i.e. $h(z)^n = f(z)$ for $z \in G$, $h$ continuous), then $h$ is holomorphic, where $f$ is a holomorphic ...
2
votes
1answer
61 views

Finding the integer solutions of the equation $3\sqrt {x + y} + 2\sqrt {8 - x} + \sqrt {6 - y} = 14$

$ 3\sqrt {x + y} + 2\sqrt {8 - x} + \sqrt {6 - y} = 14 $ . I already solved this using the Cauchy–Schwarz inequality and got $x=4$ and $y=5$. But I'm sure there is a prettier, simpler solution ...
0
votes
1answer
42 views

What is the algebra involved in solving the inequality $\sqrt x\le 2$

I would like to know how one would solve $\sqrt x\le 2$ algebraically. How do you get rid of the radical sign? Do you square both sides? Why is this allowed to do in an inequality? I already have ...
1
vote
1answer
23 views

Sign of Fractional Exponents

When calculating a number to a fractional exponent or fractional nth root, in what cases is there both a positive and negative solution as opposed to just a positive or just a negative solution?
0
votes
1answer
28 views

How to rationalize this root form?

Suppose that we have a equation like this: $$\sqrt{a+b+2\sqrt{ab}}$$ or $$\sqrt{a+b-2\sqrt{ab}}$$ In order to rationalize it, we can apply the formula: $$\sqrt{a} + \sqrt{b} = ...
3
votes
3answers
67 views

Difficulty in “expressing radical as square”

I have to get from this expression: $(4+2\sqrt3)(\sqrt{2-\sqrt3})$ To this expression: $\sqrt2+\sqrt6$ I tried to square $(4+2\sqrt3)$ and put it inside the radical, so: ...
0
votes
4answers
56 views

Prove that condition is rational

I tried to solve this about hour, but I can't... $$\begin{align} \sqrt{7+4\sqrt{3}} - \sqrt{3} \end{align}$$ Answer should be 2. I don't need to solve this for me, I just need explanation how to ...
0
votes
1answer
14 views

Rewriting the difference of two $3/2$-powers

There's this part in this problem where it goes $\frac{8}{27}\left[\left(\frac{22}{4}\right)^{3/2} - \left(\frac{13}{4}\right)^{3/2}\right]$ and it equals $\frac{22\sqrt{22} - 13\sqrt{13}}{27}$. If ...
4
votes
4answers
72 views

Rationalize $\left(\sqrt{3x+5}-\sqrt{5x+11} -\sqrt{x+9}\right)^{-1}$

I was trying to find if there a method similar to multiplying and dividing by the conjugate $$\frac{1}{\sqrt{3x+5}-\sqrt{5x+11} - \sqrt{x+9}},$$ but that doesn't seem to work here. Also, is there a ...
5
votes
4answers
101 views

Calculus Question: $\int\frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x-1}dx$

How to evaluate integral $$\int\frac{\sqrt{x^2-1}+x}{\sqrt{x^2-1}+x-1}dx?$$ I tried substitution $u^2=x^2-1$ and $u=\sqrt{x^2-1}+x$ but it turns out too complicated. Could anyone here help me to ...
2
votes
2answers
56 views

Find a positive number $\delta<2$ such that $|x−2| < \delta \implies |x^2−4| < 1$

I have to find a positive number $\delta<2$ such that $|x−2| < \delta \implies |x^2−4| < 1$. I know that $ \delta =\frac{1}{|x+2|} $ has this behaviour, but it is not guaranteed for it to ...
1
vote
1answer
53 views

How to integrate $\sqrt{1+(2/3)x}$?

How would you solve the following (step by step please!): $$\int^6_5\sqrt{1+\frac23x}\ dx$$ I started with $u=1+\frac23x$, $du=\frac23\,dx$, now what?
2
votes
4answers
58 views

How to show $\lim\limits_{x \to \infty}[x(\sqrt {x^2+a} - \sqrt {x^2+b})]=\frac{a-b}{2}$

I need to prove the result without using L'Hopitals rule $$\lim\limits_{x \to \infty}[x(\sqrt {x^2+a} - \sqrt {x^2+b})]=\frac{a-b}{2}$$ but this seems quite miraculous to me and I'm not quite sure ...
3
votes
1answer
49 views

convergence of $u_n=\sqrt{a_1+\sqrt{a_2+\sqrt{\dots+\sqrt{a_n}}}}$

Let $(a_n)_{n\in\mathbb{N}}$ be a real strictly positive sequence, and $\forall n\in\mathbb{N},u_n=\sqrt{a_1+\sqrt{a_2+\sqrt{\dots+\sqrt{a_n}}}}$ I have shown that for $a_n=1$, $u_n$ converges to ...
0
votes
1answer
34 views

How is this root simplified [closed]

As seen on wolfram Alpha. I factored out 3^2, how is the further simplification done? Particularly how and why was 2/69 gotten in the radical. ...
5
votes
4answers
350 views

General formula of repeated roots.

Prove that $$\underbrace{\sqrt{k\sqrt{k\sqrt{k\sqrt{\cdots\sqrt{k}}}}}}_{n\text { times}}=k^{1-1/2^n}$$ How do I derive this formula?
0
votes
1answer
73 views

How do you simplify this radical [closed]

$\sqrt{\frac{10}{7}}$ Please tell me how to simplify this problem. Thanks!
4
votes
3answers
117 views

Let $\alpha > 0$, use mathematical induction to prove that

Let $\alpha > 0$, use mathematical induction to prove that $$\sqrt{\alpha+\sqrt{\alpha+\sqrt{\alpha+...+\sqrt{\alpha}}}} < \frac{1+\sqrt{4\alpha+1}}{2}$$ The square root sign appears n times ...
5
votes
1answer
52 views

Rationalized limit denominator, still undefined (divide by zero), how to solve?

I am trying to solve: $$\lim_{x \to 2}\frac{\sqrt{x+2} - \sqrt{3x-2}}{\sqrt{4x+1} - \sqrt{5x-1}}$$ My first step is to multiply by the conjugate to rationalize the denominator. $$\lim_{x \to ...
-2
votes
2answers
78 views

Solving the sum of Radicals and finding the value [closed]

The value of $\sqrt{43-12\sqrt7}+\sqrt{16+6\sqrt7}$
2
votes
1answer
146 views

Simplify the radical

I need to simplify this radical, $\sqrt{2+e^{8t}+e^{-8t}}$ How is this done? I do not know where to go from here to simplify this further.