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

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7
votes
0answers
96 views

A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Using a special case of an identity by Ramanujan, we find that given the roots $x_i$ of $$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$ which, since its discriminant is negative, always has three real ...
1
vote
3answers
60 views

Integration of $1/(x\sqrt{25x^2-1})$

$$\int{\frac{1}{x\sqrt{25x^2-1}}}\,dx$$ Let $x=\frac{1}{5}u$ Now when I substite it and simplify I get $$\int{\frac{1}{u\sqrt{u^2-1}}}\, du$$ There is a trig identity which says that this is equal ...
2
votes
2answers
71 views

A reduction formula for $\int_0^1 x^n/\sqrt{9 - x^2}\,\mathrm dx$

Let $$I_n = \int_0^1 \frac{x^n}{\sqrt{9 - x^2}}\,\mathrm dx$$ Using integration, show that $$nI_n = 9(n - 1)nI_{n - 2} - 2\sqrt2$$ I've found that $\displaystyle I_0 = ...
12
votes
1answer
208 views

Something strange about $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ and its friends

We have the nice radical identity involving $d = 163$, $$-\sqrt{ 44- \sqrt{ 44 - \sqrt{ 44-x}}}=x,\quad\quad x = 2-2\sum_{n=1}^{27}\cos\left(\frac{2\pi\, t_1(n)}{163}\right)=-6.15824\dots$$ where ...
6
votes
2answers
80 views

Evaluate this Trigonometric Expression

Evaluate $$ \sqrt[3]{\cos \frac{2\pi}{7}} + \sqrt[3]{\cos \frac{4\pi}{7}} + \sqrt[3]{\cos \frac{6\pi}{7}}$$ I found the following $\large{\cos \frac{2\pi}{7}+\cos \frac{4\pi}{7} + \cos ...
0
votes
0answers
27 views

Noetherian ring and radical

Show that in a Noetherian ring $I$ and $J$ have the same radical if and only if there is a positive integer $N$ such that $I^N \subset J$ and $J^N \subset I$. [Hint: for the ``if'' direction, use a ...
0
votes
0answers
31 views

noetherian Ring

Let $I$ be an ideal in a Noetherian ring $R$. Prove that there exists a positive integer $N$ such that $(rad(I))N ⊂ I$. [Hint:Let $rad(I)=⟨g_1,...,g_k⟩$,and suppose $g_i^{n_i} ∈I$.Use ...
6
votes
4answers
103 views

How to solve $x^{2/3}=4$?

OK I know this sounds pretty stupid, but I am stuck on solving $x^{{2}/{3}}=4$. I rewrote it to $\sqrt[3]{x^2}=4$, but I don't know what to do next. Would the radical go away if I took the ...
0
votes
0answers
34 views

Radical Ideal for algebra

Show the following: a) rad(IJ)=rad(I∩J)=rad(I)∩rad(J) b) rad(I)=R if and only if I=R c) if P is prime rad(P^n)= P for all n d) Let F be a field and T a subset of F^n . Show that the ideal I(T) ...
0
votes
2answers
36 views

How to transform a fraction with square root in denominator?

Simple question but my math is a bit rusty. How does $\frac{20}{\sqrt5} = 4\sqrt5$? I tried multiplying by $\frac{\sqrt5}{\sqrt5}$ but that's not right.
0
votes
4answers
90 views

How to find the value of $\ \lim_{x\to 3^+}\frac{\sqrt{x^2-9}}{x-3}$?

How do I find the value of this limit? $$\lim_{x\to 3^+}\frac{\sqrt{x^2-9}}{x-3}$$ It says that it's approaching from right side to 3 right? I tried subsitituting the 3 into the variables, and got ...
6
votes
1answer
148 views

How to evaluate the integral of $\sqrt{\sin\sqrt x}\cos \sqrt x / ( 1+x^2)$?

$$ \int \frac{\sqrt{\sin\sqrt x}\cos \sqrt x}{1+x^2} dx $$ I have tried combinations of $x=t^2$, integration by parts, $\tan\left(\dfrac u2\right)$ substitutions it got even more complicated. Is ...
1
vote
1answer
27 views

Evaluating a decimal exponent

The example I am stuck with is $(-1)^.456$. According to my calculator, the answer is -1. According to WolframAlpha and the Google Calculator, it is a complex number. WolframAlpha -- ...
1
vote
1answer
27 views

How to solve implicit differentiation with radicals?

Here is the question: $$\sqrt{y^2\sin^2x + x^2\cos^2 x} = 4xy$$ I know about product rule and such but I'm exactly sure how to begin. edit: I'm just trying to get the first derivative of this ...
8
votes
10answers
170 views

How to evaluate $\lim_{n\to\infty}\sqrt[n]{\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}}$

Im tempted to say that the limit of this sequence is 1 because infinite root of infinite number is close to 1 but maybe Im mising here something? What will be inside the root? This is the sequence: ...
2
votes
2answers
33 views

Calculate the following intergral

I have to calculate the following integral $$ \int \sqrt[3]{1+x\ln{x}} * (1+\ln{x}) dx$$ I have thought about using the following notation: $$ t = {1+x\ln(x)} => x\ln{x} = t-1 $$ But here I ...
0
votes
4answers
53 views

Inequality $\sqrt{x^2-16} < 2-x$. [closed]

$\sqrt{x^2-16}<\:2-x$ How can I resolve this inequality ?
1
vote
0answers
37 views

Solvability by radicals of Polynomials defined by a recurrence relation

I want to determine the smallest integer $m$ such that the polynomial $P_{n}(x)$, $n\geq m$, given by : $$\left \lbrace \begin{array}{l} P_{n+1}(x) = P_n(x) (x-n-1) + \prod\limits_{i = 0}^n x-i\\ ...
1
vote
1answer
23 views

Simplifying this Complex Radical: $-{1/5}(x-4)^{5/3} - 2(x-4)^{2/3}$

$-{1/5}(x-4)^{5/3} - 2(x-4)^{2/3}$ I need to get this function into a simpler form so that I can analyze it's domain, limits, derivative and second derivative more easily. I am very bad with radicals ...
0
votes
1answer
51 views

Evaluating the limit $\lim_{x\to-1}\frac{\sqrt{x}-1}{x-1}$

How would you solve $\displaystyle\lim_{x\rightarrow-1}\left(\dfrac{\sqrt{x}-1}{x-1}\right)$ ? I tried multiplying it by the conjugate. I don't know how to get rid of the square root.
2
votes
3answers
81 views

Convergence of $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}\Big(\sqrt[3]{(n+1)^{2}} - \sqrt[3]{n^{2}}\Big)$

$\displaystyle\sum_{n=1}^{\infty}\frac{\sqrt[3]{(n+1)^{2}} - \sqrt[3]{n^{2}}}{n}$ Converging or Diverging? I guess I have to lower the fraction so that the roots will get away and I will have ...
0
votes
2answers
53 views

Why can you not cancel-down the square within a square-root?

$y = \sqrt{x²+5x+6.25} \iff y = \sqrt{(x+2.5)²} $ but $\iff y = x+2.5$ together with $y = -x-2.5$ is wrong. Why? What are the underlying rules?
3
votes
2answers
54 views

Limit of square root function at $x \to 6$

I'm trying to find the limit of the following function at $x \to 6$: $$\frac{x^2-36}{\sqrt{x^2-12x+36}}$$ i've simplified it so that it becomes $\dfrac{(x+6)(x-6)}{\sqrt{(x-6)^2}}$, which simplifies ...
2
votes
4answers
61 views

An Impossible Ratio

I'm facing a bit of a difficulty thinking about the aspect ratio of A4 paper. The beauty of this paper size is that when it is folded in half along the longer side, it becomes A5 paper which has ...
0
votes
1answer
61 views

Nth root of n is greater than 1?

A proof I did recently called upon a "fact" which my prof called without giving explanation or proof, which is the "fact" that $\sqrt[n]{n}>1$, how can this be shown?
1
vote
3answers
56 views

Express this sum of radicals as an integer?

I have read somewhere that the radical $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$ and I don't understand it. How do you solve this(when the RHS is unknown)?
2
votes
1answer
34 views

Show that the radical of the ideal is equal $ \langle X,Y\rangle $

$\def\Rad{\operatorname{Rad}}$ Could you give me some hints how I can solve the followig exercise? Show that the $\Rad(I)$ of the ideal $I=\langle X^5,Y^3\rangle $ of the ring $\mathbb{C}[X,Y]$ is ...
5
votes
6answers
136 views

How would you show that $\sqrt{14+4\sqrt{10}} - \sqrt{14-4\sqrt{10}} = 4$?

I've recently seen a Highschool problem and I was wondering, how would you show that $$\sqrt{14+4\sqrt{10}} - \sqrt{14-4\sqrt{10}} = 4$$ Thank you for your time,
0
votes
2answers
36 views

$\sqrt{1\pm10\varepsilon+\varepsilon^2}=1\pm P(\varepsilon)$. Is there a better way than mine to find $P(\varepsilon)$?

Some days ago we did a classwork, and there was this exercise: Using the limit definition, verify $$\displaystyle \lim_{x\to0} \frac{3x^2-1}{x+1}=-1.$$ From $\displaystyle ...
1
vote
3answers
70 views

Proving $\sum^{n}_{k=1} \frac{1}{\sqrt{k}}>\sqrt{n}$ by induction

Prove that $$\sum^{n}_{k=1} \frac{1}{\sqrt{k}}>\sqrt{n}$$ for all $n\in \mathbb{N}$ where $n\geq2$. I've already proven the base case for $n=2$, but I don't know how to make the next step. Is the ...
2
votes
2answers
33 views

Limit Computation, Sandwich.

I have the following question. I was asked to compute the following limit: Let $A_1 ... A_k$ be positive numbers, does exist: $$ \lim_{n \rightarrow \infty} (A_1^n + ... A_k^n)^{1/n} $$ My work: ...
3
votes
3answers
125 views

How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?

Prove that for any positive integer $n$, $$2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$$ Progress I think Riemann sum should be used for ...
0
votes
1answer
70 views

What does 8 root 2.5 ($\sqrt[8]{2.5}$) mean?

$8$ root of $2.5$? I have calculated the equation and found $12.6$ is the answer. However, the answer given is $1.12$ so what could I have done wrong? Thanks
3
votes
2answers
293 views

Simplifying radicals

I am stuck in the following puzzle and couldn't find a way to approach this. $\sqrt{5 + \sqrt{5} + \sqrt{3 + \sqrt{5} + \sqrt{14 + \sqrt{180}}}}$ Please help.
1
vote
4answers
63 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
99 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.
2
votes
2answers
62 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
36 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
43 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
33 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
28 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?
0
votes
3answers
58 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
43 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
74 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
87 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
95 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
53 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
73 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
273 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 $\sqrt{x}$ ...