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

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6
votes
1answer
76 views

Finding $\sqrt{17}$ and $\sqrt{257}$ in the regular $17$-gon and $257$-gon?

Given the regular $n$-gon formed by the $n$-th roots of unity. For some $n$, how do we find $\sqrt{n}$ using the sum/difference of line segments? $n=5:$ It is enough to use one line segment: If ...
0
votes
2answers
54 views

Evaluate the definite integral $\int_{0}^{a}\frac{dx}{(a^2+x^2)^{3/2}}$

I'm trying to solve this integral with trigonometric substitution but am having a ton of trouble: $$\int\limits_{0}^{a}{\frac{dx}{(a^2+x^2)^{\frac{3}{2}}}}$$ I tried $x=a\tan{\theta}$ and thus ...
2
votes
1answer
28 views

$n$th root of $x$ - technical term for $n$?

As you can see in the title, I want to know how the number before a root is called. For example, if you have the cubic root of 8, I want to know how the 3 before the roof is called. Actually, I ...
6
votes
4answers
147 views

How many values does $\sqrt{\sqrt{i}}$ have?

Wolfram says, there are only two roots, but $\sqrt{i}$ already gives two roots. So if we express them in Cartesian form we can take square roots of them separately and end up with four roots. ...
0
votes
2answers
25 views

Question on double inequality with radicals

A really simple question, but I thought I'd ask anyway. Does $n<x^n<(n+1)$ imply $\sqrt[n] n < x < \sqrt[n] {n+1}$? Thank you very much.
6
votes
3answers
65 views

Finding the square root of $6-4\sqrt{2}$

I found this standupmaths video on YouTube about the A4 paper puzzle. I really liked the puzzle and managed to get the answer by using a calculator. However, the answer (which I won't spoil), led me ...
4
votes
5answers
65 views

Why is the solution to $\sqrt{6-5x}=x$ only $x=1$ and not $x=-6$? [duplicate]

I solved the equation $\sqrt{6-5x}=x$ as follows: $$(\sqrt{6-5x})^2=x^2$$ $$6-5x=x^2$$ $$0=x^2+5x-6=(x+6)(x-1)$$ $$x=-6 \quad \text{or} \quad x=1$$ If I plug in $x=-6$ into the original equation, I ...
1
vote
3answers
62 views

Problematic square root [duplicate]

Ok, here is what I think. Please correct me if I am wrong. $$\sqrt{9} \neq 3$$ and also $$\sqrt{9} \neq -3$$ Now let's assume, that above statements are false, then we have $-3 = \sqrt{9} = 3$ and ...
0
votes
1answer
72 views

Solution to the equation $x^3-3=2\sqrt{x+2}$

Solve the equation $x^3-3=2\sqrt{x+2}$. I have tried to let $t=\sqrt{x+2}$ then we have $$\begin{cases} x^3-3&=2t \tag 1\\ t^2 &=x+2 \end{cases}$$ But I've stuck here... Any help ...
2
votes
1answer
40 views

Integral of a trig function divided by the square root of a polynomial: $\int_a^b\frac{\sin x}{\sqrt{(x-a)(b-x)}}dx$?

I was trying to help some physics students with an integral on their homework and they've presented me with something that has me stumped. The integral they are working on is: $$\int_a^b\frac{\sin ...
0
votes
3answers
48 views

Why is $(\sqrt{x^2})$ equal to $|x|$ [duplicate]

I don't understand why you have to write the absolute value sign when solving for the square root of $x$ squared. Shouldn't the answer automatically be positive? Why is the absolute value sign ...
1
vote
2answers
42 views

Use the Intermediate Value Theorem to prove that $\sqrt s$ exists?

I need to prove with the Intermediate Value Theorem that $\sqrt s$ exists, where $s > 0$. My textbook states this definition of the Intermediate Value Theorem: Suppose that $f$ is continuous on ...
11
votes
2answers
181 views

How do I calculate this limit: $\lim\limits_{n\to\infty}1+\sqrt[2]{2+\sqrt[3]{3+\dotsb+\sqrt[n]n}}$?

I have seen this question on the internet and was interested to know the answer. Here it is : Calculate $\lim\limits_{n\to\infty}(1+\sqrt[2]{2+\sqrt[3]{3+\dotsb+\sqrt[n]n}})$? Edit : I really tried ...
3
votes
2answers
117 views

Does $\overline{ \sqrt{1 + i}} = \sqrt{1-i} \ $?

I am having trouble with complex conjugates today. Can someone help me? $$\overline{ \sqrt{1 + i}} \stackrel{\color{#2222FF}{?}}{=} \sqrt{1-i} \tag{$\ast$} $$ In this case, since $\cos ...
4
votes
5answers
109 views

Calculate $\lim_{n\to\infty}(\sqrt{n^2+n}-n)$. [duplicate]

Introduction: An exercise from "Principles of mathematical Analysis, third edition" by Rudin, page 78. Exercise: Calculate $\lim_{n\to\infty}(\sqrt{n^2+n}-n)$. Explanation: I have a hard ...
1
vote
2answers
72 views

Evaluating limit $\lim_{n\to\infty}({\sqrt{4^n + 3^n} - 2^n})$

I have to find: $$\lim_{n\to\infty}\left({\sqrt{4^n + 3^n} - 2^n}\right)$$ I plugged in some numbers and it seems as if this sequence were approaching infinity, but I do not know how to begin ...
1
vote
2answers
77 views

How to solve $\sqrt {35 - 5i}$ [duplicate]

Need some hints on how to Solve $\sqrt {35 - 5i}$ Attempt. I factorized 5 out and it became $\sqrt {5(7-i)}$ I just want to know if it can be solved further. Thanks.
5
votes
3answers
274 views

Finding $\lim_{x\to\infty}\frac{\sqrt{x+\sqrt{x+\sqrt{x}}}}{x}$

I saw some resolutions here like $\sqrt{x+\sqrt{x+\sqrt{x}}}- \sqrt{x}$, but I couldn't get the point to find $\lim_{x\to\infty}\frac{\sqrt{x+\sqrt{x+\sqrt{x}}}}{x}$. I tried ...
0
votes
3answers
29 views

Squaring a binominal

I have a really simple question that I can't find the answer to. In a algebra test, I was asked to simplify $(5 + $$\sqrt{3}$)$^2$. What I did was to square each term individually: ...
2
votes
2answers
54 views

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$. I begin by letting $n=1$ then $\frac{1}{2}<\frac{1}{\sqrt{3}}$. Then assume $\frac{1\cdot ...
0
votes
0answers
23 views

$x\in\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$, s.t. $x^2=p+q\sqrt{30}$

Does there exists $x\in\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ of the form $$ x=a+b\sqrt{2}+c\sqrt{3}+d\sqrt{5}+e\sqrt{6}+f\sqrt{10}+g\sqrt{15}+h\sqrt{30},\tag{1} $$ where ...
2
votes
2answers
31 views

Solving a radical equation with trinomials on both sides

$$8\sqrt{a^2-4a-16}=3a^2-12a-64$$ I do know the standard procedure—square both sides, isolate square root, square again, check solutions to make sure they are real, etc. However, for a problem such ...
7
votes
7answers
375 views

How do I simplify and evaluate the limit of $(\sqrt x - 1)/(\sqrt[3] x - 1)$ as $x\to 1$?

Consider this limit: $$ \lim_{x \to 1} \frac{\sqrt x - 1}{ \sqrt[3] x - 1} $$ The answer is given to be 2 in the textbook. Our math professor skipped this question telling us it is not in our ...
4
votes
4answers
46 views

Evaluate the limit of function $\lim_{x\to\infty}\frac{(9x^2+1)^{1/2}}{x+2}.$

Find the limit: $$\lim_{x\to\infty}\frac{(9x^2+1)^{1/2}}{x+2}.$$ I want to divide each of the terms by the highest power of $x$ but I failed to elimite the square root on it.
2
votes
1answer
60 views

Roots of $x^3-x+1$

I am trying to find nice explicit formulas for the roots of the polynomial $x^3-x+1$. Is there some clever way to write down the roots in a reasonably easy way? I found the roots, but my expressions ...
1
vote
1answer
46 views

Integral with simple fractions

I have a problem with this integral $$\int_\ \frac{\cos x }{\sin x \sqrt{1+\cos^2x}} \, dx$$ Using substitution $u = \sin x $ we get $$\int_\ \frac{1 }{\ u \sqrt{2-u^2}} \, du$$ I think the ...
8
votes
8answers
242 views

Calculating $\lim_{n\to\infty}\sqrt[n]{ \sqrt[n]{n} - 1 }$

I know that $$\lim_{n\to\infty} \sqrt[n]{ \sqrt[n]{n} - 1 } = 1,$$ but I'm unable to prove it. I could easily estimate that it's at most $1$, but my best estimation from below is that the limit ...
37
votes
12answers
5k views

Why do I get one extra wrong solution?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by (-1): $$\sqrt{x}=x-2$$ power of 2: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $1$ is not a ...
3
votes
4answers
91 views

How to show $\frac{1}{2\sqrt{2} + \sqrt{3}} = \frac{2\sqrt{2} - \sqrt{3}}{5}$?

Show that: $$ \dfrac{1}{2\sqrt2+\sqrt3}=\dfrac{2\sqrt2-\sqrt3}{5}$$ So I multiplied everything by $\sqrt3$ Then I got $$\frac{\sqrt{3}}{2\sqrt{2}+3}$$ Then multiply it by $\sqrt2$ to obtain ...
3
votes
4answers
303 views

Solving this limit $\lim_{x\to 3}\frac{x^2-9-3+\sqrt{x+6}}{x^2-9}$.

The question is $\lim_\limits{x\to 3}\frac{x^2-9-3+\sqrt{x+6}}{x^2-9}$. I hope you guys understand why I have written the numerator like that. So my progress is nothing but ...
0
votes
1answer
46 views

Having issue with tan(7pi/6) converting degrees then to radical

I'm trying to work out the following problem: tan 7pi / 6 I first converted radians to ...
11
votes
6answers
152 views

Simplifying nested square roots ($\sqrt{6-4\sqrt{2}} + \sqrt{2}$)

I guess I learned it many years ago at school, but I must have forgotten it. From a geometry puzzle I got to the solution $\sqrt{6-4\sqrt{2}} + \sqrt{2}$ My calculator tells me that (within its ...
7
votes
8answers
499 views

Convergence of $\sin{\pi\sqrt{n}}$

Revising for an exam: Let $a_n = \sin{(\pi\sqrt{n})}.$ Show that: (i) $a_{n+1} - a_{n} \rightarrow 0$ (ii) The sequence $(a_n)$ is bounded. (iii) $(a_n)$ does not converge. My ...
1
vote
1answer
25 views

Asymptotic bound with square-roots

Let $f(n)$ and $g(n)$ be two increasing functions of $n$ such that: $$ f \leq g + O(\sqrt{g}) + O(\sqrt{f}) $$ Is it true that: $$ f \leq g + O(\sqrt{g}) $$ ? If not, then what would be a good ...
2
votes
3answers
108 views

limit of nth root n is e?? where's the mistake

$$\lim_{n\to \infty} \sqrt[n] n = \lim_{n\to \infty} n^{\frac{1}{n}} = \lim_{n \to \infty} \{(1+(n-1))^{\frac{1}{n-1}}\}{^{(n-1)\frac{1}{n}}} = \lim_{n\to \infty} e^{\frac{n-1}{n}} = e$$ But this ...
6
votes
4answers
170 views

Why is $\sqrt {12} = 2 \sqrt 3$?

Why $\sqrt {12} = 2 \sqrt 3$? It is obvious? If we considered the function $f(s) = s^2 $ it is injective on positive numbers so we obtain the conclusion. But in the same time it is an equality ...
0
votes
4answers
401 views

Trick to solve this limit $\lim_{x \to\infty} \sqrt{x^2+3x}-x$ [closed]

The question is $$\lim_{x \to\infty} \sqrt{x^2+3x}-x$$ I divided and multiplied by $x^2$ but it gave me answer $0\cdot\infty$ which is undefined I suppose. any hint !! (PS: avoid using L Hospital's ...
-2
votes
1answer
27 views

Procedure about simplifying radicals

I have this: $$\frac {(2n+1)^3-8n(n^2-1)}{\sqrt{n(n+1)(n+2)(n+3)}}=\sqrt{\frac{144n^4+336n^3+220n^2+28n+1}{n(n+1)(n+2)(n+3)}}$$ What does this mean? What's the step by step procedure to do this? ...
2
votes
5answers
77 views

Limit of: $\lim_{n \to \infty}(\sqrt[3]{n^3+\sqrt{n}}-\sqrt[3]{n^3-1})\cdot \sqrt{(3n^3+1)}$

I want to find the limit of: $$\lim_{n \to \infty}(\sqrt[3]{n^3+\sqrt{n}}-\sqrt[3]{n^3-1})\cdot \sqrt{(3n^3+1)}$$ I tried expanding it by $$ ...
4
votes
4answers
142 views

Find the value of the infinite product $\sqrt\frac12\cdot\sqrt{\frac12+\sqrt\frac12}\cdot\sqrt{\frac12+\sqrt{\frac12+\sqrt\frac12}}\cdots$

Find $\sqrt{\frac{1}{2}}.\sqrt{\frac{1}{2}+\sqrt\frac{1}{2}}.\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+\sqrt\frac{1}{2}}}....\infty$ Let ...
4
votes
1answer
87 views

Square and cubic roots in $\mathbb Q(\sqrt n)$

Here is my question : Let $n$ a squarefree positive integer, $m \ge 2$ an integer and $a+b \sqrt n \in\mathbb Q (\sqrt n).$ What (sufficient or necessary) conditions should $a$ and $b$ satisfy so ...
4
votes
3answers
57 views

How to solve the following arithmetic radical problem?

$$ 2(4\sqrt{7} + 1 + 3\sqrt{7} + 2) $$ I distribute first right? $$ 8\sqrt{14} + 2 + 6\sqrt{14} + 4$$ $$ 14\sqrt{14} + 6$$ BUT IT LOOKS LIKE ITS SUPPOSED TO BE $$14\sqrt{7} + 6$$ I also have a ...
3
votes
2answers
70 views

Why do I get an imaginary result for the cube root of a negative number?

I have a function that includes the phrase $(-x)^{1/3}$. It seems like this should always evaluate to $-(x^{1/3})$. For example, $-1 \cdot -1 \cdot -1 = -1$, so it seems that $(-1)^{1/3}$ should equal ...
2
votes
0answers
66 views

Finding a limit with negative infinity (Square root)

I'm given this question $$\lim_{x\rightarrow -\infty }\left(\sqrt{x}-\frac{2+x}{\sqrt{x}}\right) $$ My attempt, $\lim_{x\rightarrow -\infty }(\sqrt{x}-\frac{2+x}{\sqrt{x}})=\lim_{x\rightarrow ...
-1
votes
2answers
80 views

Prove that limit of the fractional part of $\sqrt{n^2+n}$ is $\frac{1}{2}$ [duplicate]

Prove that $$\operatorname{frac}(\sqrt{n^2 + n}) \to \frac{1}{2}$$ ($n \in \mathbb{N}$, $\operatorname{frac}$ is fractional part of number) I think I should use just definition of limit and find $N$ ...
1
vote
2answers
72 views

solving $\int \frac{dx}{\sqrt{-x^2-12x+28}}$

$$\int \frac{dx}{\sqrt{-x^2-12x+28}}$$ First we need to use completing the square $-(x^2+12x-28)=-(x+6)^2+64$ So we have $\int \frac{dx}{\sqrt{-(x+6)^2+64}}$ I know that it is a general form of ...
2
votes
2answers
108 views

Integrate the square root of the ratio of two quadratic polynomials

$$\int \sqrt{\frac{x^2+x-1}{x^2-1}} dx$$ I have been trying to find this integral for a while and I just can't. Does it even have a closed form?
28
votes
1answer
640 views

Prove that $a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$

Let $a$, $b$ and $c$ be non-negative numbers. Prove that: $$a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}.$$ I have a proof, but my proof is very ugly: it's ...
0
votes
5answers
225 views

How to solve the integral $\int\frac{x-1}{\sqrt{ x^2-2x}}dx $

How to calculate $$\int\frac{x-1}{\sqrt{ x^2-2x}}dx $$ I have no idea how to calculate it. Please help.
3
votes
4answers
74 views

All Solutions for $(-256)^{\frac{1}{4}}$ and $1^{\frac{1}{5}}$ Imaginary Roots?

This is a question about the imaginary roots of the two equations $$ (-256)^{\frac{1}{4}} \qquad\text{and}\qquad 1^{\frac{1}{5}}. $$ For the first one I've worked out that 2 of the solutions are ...