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

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0
votes
3answers
55 views

What's wrong about this limit?

i've got a very simple exercise of limit, and i started solving it, but it won't get out of indetermination. The limit is $$\lim_{x\to 1} = \frac{x^2-\sqrt x}{1-\sqrt x}$$ If i use L'Hôpital's it ...
0
votes
0answers
19 views

Proving that if $m,n,p,q\in\mathbb{Z^+}, \sqrt[p]{m}\in\mathbb{R}\setminus\mathbb{Q}$ then $\sqrt[p]{m}+\sqrt[q]{n}\in\mathbb{R}\setminus\mathbb{Q}$

If $\sqrt{m}+\sqrt[q]{n}=r$ rational, the rationality of $\sqrt{m}$ is derived expanding $(r-\sqrt{m})^q$ using the binomial theorem: after rearrangement, isolating the terms containing odd powers of ...
9
votes
5answers
2k views

Unexpected result from Euler's formula

I am a bit confused with a result I get from Euler's formula: $e^{2\pi i} = 1$ $\sqrt[3] { e^{2\pi i} }= \sqrt[3]{ 1 }$ $(e^{2\pi i})^{\frac{1}{3}}= 1$ $e^{\frac{2}{3} \pi i} = 1$ This last ...
0
votes
3answers
62 views

Multiplication of real and complex radicals

If I have, for example, the product $\sqrt{7+\sqrt{22}}\sqrt[3]{38+i\sqrt{6}} $ Can I perform the multiplication or this cannot be done and only remains to leave the product in this form?
7
votes
3answers
938 views

What am I doing wrong in calculating the following limit?

$$\lim_{x\to-2} \frac{x+2}{\sqrt{6+x}-2}=\lim_{x\to-2} \frac{1+2/x}{\sqrt{(6/x^2)+(1/x)}-2/x^2}$$ Dividing numerator and denominator by $x \neq0$ ...
1
vote
2answers
69 views

Prove that $(√5 - 1)/2$ is irrational.

Please help me prove that $(√5 - 1)/2$ is irrational. I know how to prove √5 is irrational: Assume that √5 is rational meaning √5 = $p/q$ $p,q$ $are$ $Z$ $and$ $q≠0$ $p^2/q^2 = 5$ $q^2 = ...
4
votes
3answers
67 views
+50

General Principles of Solving Radical Equations

What are the general ways to solve radical equations similar to questions like $\sqrt{x+1}+\sqrt{x-1}-\sqrt{x^2 -1}=x$ ...
0
votes
2answers
75 views

Solve the simultaneous equations for real numbers $x$ and $y$: $ \sqrt{x+a} + \sqrt{x-a} = 3 $ and $ x+y=5 $

Question: Let $a$ be a real number. Solve the simultaneous equations for real numbers $x$ and $y$: $$ \sqrt{x+a} + \sqrt{x-a} = 3 $$ $$ x+y=5 $$ My attempt: Consider ...
4
votes
3answers
101 views

Why is $5\tan(54^\circ) = \sqrt{25 + 10\sqrt{5}}$ and $\tan\left(\frac{\pi}{5}\right) = \sqrt{5 - 2\sqrt{5}}$?

On the Wikipedia Page about Pentagons, I noticed a statement in their work saying that $\sqrt{25+10\sqrt{5}}=5\tan(54^{\circ})$ and $\sqrt{5-2\sqrt{5}}=\tan(\frac {\pi}{5})$ My question is: How would ...
5
votes
2answers
75 views

Radical equation $\sqrt{x+1}+\sqrt{x-1}-\sqrt{x^2 -1}=x$

The Question: $\sqrt{x+1}+\sqrt{x-1}-\sqrt{x^2 -1}=x$ Only thing I can take from this is that $x^2 -1=(x+1)(x-1)$, but I don't think that would help in any way. I know the answer, but I don't know ...
3
votes
3answers
56 views

express the value of an expression as a common factor

In the following problem, by adding $0.141414$..., $0.414141..., 0.151515...$, and $0.515151...$, I get $1.111....$ Then the expression becomes square root ($11 \times 1.1111$....). My answer is $11 ...
1
vote
1answer
53 views

Is the answer key wrong? or it's me?

ok, so im reviewing for a math test and the following question is from the practice final exam. Rationalize the denominator in the example: $$\frac{\sqrt {2}}{\sqrt {x-3}}$$ after multiplying both ...
3
votes
5answers
136 views

How to prove $\sqrt3 + \sqrt[3]{2}$ is a irrational number?

This is an exercise from R. Courant's book: How to prove $\sqrt3 + \sqrt[3]{2}$ is a irrational number? The solution is to construct a equation to prove, but is there any other method to prove this, ...
3
votes
1answer
98 views

Close approximation for absolute value function

I made a very acurate approximation function for $\sqrt{n^{2}+1}$ It is $\sqrt{n^{2}+1}\approx\frac{2n(n^{2}+1)}{2n^{2}+1}+\frac{2n^{2}+1}{n(4(2n^{2}+1)^{2}+1)}$ From this I can make a very close ...
3
votes
2answers
213 views

a general continued fraction satisfying $\frac{(i+\Theta\sqrt{z})^m}{(i-\Theta\sqrt{z})^m}=\frac{(ik+\sqrt{z})^{m+1}}{(ik-\sqrt{z})^{m+1}}$

Given any natural number $m\gt2$, let $z$,$k$ be complex numbers, where $i=\sqrt{-1}$ and consider the general continued fraction $$\Theta(k,z,m)=\cfrac{(m+1)}{km+\cfrac{z(0m-1)(2m+1)} ...
1
vote
1answer
49 views

Why is the square root of a number not plus or minus? [duplicate]

For example, $\sqrt{4}$. I've asked a bunch of people and I get mixed answers all the time, as to whether it is $-2$ and $+2$ or just $+2$. How about if there's a negative in front of the square root ...
3
votes
2answers
194 views

A pair of continued fractions that are algebraic numbers and related to $a^2+b^2=c^m$

Similar to the cfracs in this post, define the two complementary continued fractions, $$x=\cfrac{-(m+1)}{km\color{blue}+\cfrac{(-1)(2m+1)} {3km\color{blue}+\cfrac{(m-1)(3m+1)}{5km\color{blue} ...
2
votes
1answer
26 views

What is the minimum value of a radical sum?

How would you find the minimum value of $\sqrt {x^2+16} + \sqrt {x^2-12x+37}$ through algebraic manipulation? Graphing will clearly show that the minimum value is 4.8, but how do you get the answer ...
1
vote
3answers
62 views

Computing $ \mathbf A^{-1/2}$, where $ \mathbf A$ is a Diagonal Matrix.

I have the following matrix : $$ \mathbf A =\begin{bmatrix} 100 & 0 \\ 0 & 1 \\ \end{bmatrix}$$ I have to compute $ \mathbf A^{-1/2}$. So I need spectral decomposition, ...
0
votes
0answers
26 views

Periodicity of the continued fraction of a square root

Writing $\sqrt{n}=[a_0; a_1, a_2, \dots ]$, at which $a_i$ does the period start? Is it $a_1$? I just put "for some $n\ge 1$, where $a_{n-1}=a_i$", is that a good enough answer?
-1
votes
2answers
77 views

If $x = 5- \sqrt{21}$, find the value of $\dfrac {\sqrt x}{\sqrt{32-2x} - \sqrt{21}}$.

PROBLEM: If $x = 5- \sqrt{21}$, find the value of $\dfrac {\sqrt x}{\sqrt{32-2x} - \sqrt{21}}$. Solution: $$x = 5- \sqrt{21}$$ $$\sqrt x = \sqrt {5- \sqrt{21}}$$ I am unable to continue from here. ...
9
votes
5answers
119 views

Find the limit of $\frac{(n+1)^\sqrt{n+1}}{n^\sqrt{n}}$.

Find $$\lim_{n\to \infty}\frac{(n+1)^\sqrt{n+1}}{n^\sqrt{n}}$$ First I tried by taking $\ln y_n=\ln \frac{(n+1)^\sqrt{n+1}}{n^\sqrt{n}}=\sqrt{n+1}\ln(n+1)-\sqrt{n}\ln(n),$ which dose not seems to ...
1
vote
1answer
23 views

Find when the product would be an integer

The problem: The sequence $\{a_n\}$ is defined recursively by $a_0=1,a_1=\sqrt[19]{2}$ and $a_n=a_{n-1}a_{n-2}^2$ for $n \geq 2$. What is the smallest positive integer $k$ such that the product ...
3
votes
1answer
74 views

Find real roots of the equation

Find all real solutions to $$\dfrac{\sqrt{x+1}}{2+\sqrt{2-x}} - \dfrac{\sqrt{x^2-x+2}}{2+\sqrt{-x^2+x+1}} = x^3-x^2-x+1$$ This question is very similar to one of my previous problem, ...
2
votes
1answer
47 views

Solving a mixed radical and quadratic equation

Solve for $x \in \mathbb{R}$ $$4x^2(x+2) +3(2x^2-4x-3)\sqrt{4x+3} +6x = 0$$ I tried taking square by isolating the radical, but the resultant equation couldn't be solved. Any help ...
10
votes
1answer
145 views

Finding all real roots of the equation $(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$

Find all real roots of the equation $$(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$$ I tried squaring the equation, but the degree of the equation became too high and unmanageable. I ...
5
votes
1answer
96 views

Solve $ 1 + \dfrac{\sqrt{x+3}}{1+\sqrt{1-x}} = x + \dfrac{\sqrt{2x+2}}{1+\sqrt{2-2x}} $

Solve for $x \in \mathbb{R}$ $$ 1 + \dfrac{\sqrt{x+3}}{1+\sqrt{1-x}} = x + \dfrac{\sqrt{2x+2}}{1+\sqrt{2-2x}} $$ I tried some substitutions and squaring but that didn't help. I also ...
5
votes
1answer
101 views

Solving a radical equation for real $x$

Solve for $x \in \mathbb{R}$ $$\dfrac{\sqrt{x^2-x+2}}{1+\sqrt{-x^2+x+2}} - \dfrac{\sqrt{x^2+x}}{1+\sqrt{-x^2-x+4}} = x^2-1$$ I tried squaring the equation but it became a sixteen degree ...
-2
votes
0answers
16 views

how can I separate multiple radicals.

I have an expression. $E_2 = \sqrt{J+\sqrt{K}}$ I want to write this in another form. Let $E_3 = \sqrt{A} + \sqrt{B}.$ Set $E_2 = E_3$. Assuming that $A > B$, we have $A = ...
2
votes
0answers
65 views

Simplify and Denest $\sqrt[3]{2+\sqrt{3}+\sqrt[3]{4}}$

I'm not too sure how to begin in denesting $\sqrt[3]{2+\sqrt{3}+\sqrt[3]{4}}$. I thought about setting it equal to $a+b\sqrt{3}+c\sqrt[3]{4}$, but that sounds a bit sketchy.
0
votes
1answer
16 views

Find elements of a set that divide an expression.

I have to determine the elements of the following set: $A = \{x\in\ \mathbb Z \vert \sqrt[3]{\frac {7x + 2}{x+5}} \in \mathbb Z \}$ I know that $x+5 \not=0$ and $x+5$ must divide $7x + 2$ but I ...
1
vote
0answers
23 views

Denesting Radicals with two different radicands

After thinking some time about denesting radicals, I wondered if it was possible to denest a radical in the form $\sqrt[a]{\sqrt[b]{\alpha}+\sqrt[c]{\beta}}$ I thought about rewriting the inside to a ...
2
votes
1answer
180 views

Use the simple continued fraction of $\sqrt{27323}$ to factor $27323$…

Use the simple continued fraction of $\sqrt{27323}$ to factor $27323$. So far I have: $\sqrt{27323} = 1 + (\sqrt{27323} - 1)$ which gives... $= 1 + \frac{1}{(\frac{1}{164.2967029})}$ I'm ...
0
votes
1answer
45 views

Why does squaring an expression with 2 subtracting terms work?

This expression can be simplified as: $$\sqrt{(x-\frac32)^2} = x - \frac32$$ Even though: $$k^2 = m^2 + n^2 \to \sqrt{k^2} = \sqrt{m^2 + n^2} \to k = \pm\sqrt{m^2 + n^2}$$ You can not remove the ...
5
votes
3answers
103 views

Evaluate the $\lim_{x \to \ -\infty} (x + \sqrt{x^2 + 2x})$

Evaluate : $$\lim_{x \to \ -\infty} (x + \sqrt{x^2 + 2x})$$ I've tried some basic algebraic manipulation to get it into a form where I can apply L'Hopital's Rule, but it's still going to be ...
1
vote
2answers
83 views

How do I solve for $m$ and $n$

While reading about nested radicals, I came across a theorem that said $\sqrt{m\sqrt[3]{4m-8n}+n\sqrt[3]{4m+n}}=\pm\frac ...
0
votes
3answers
58 views

Why does this Calculus II sequence diverge?

What steps should I take to understand why this sequence diverges (specifically, not to +∞ or -∞)?
4
votes
3answers
91 views

Can't seem to solve a radical equation? Question is : $\sqrt{x+19} + \sqrt{x-2} = 7$

So there is this equation that I've been trying to solve but keep having trouble with. The unit is about solving Radical equations and the question says Solve: $$\sqrt{x+19} + \sqrt{x-2} = 7$$ I ...
4
votes
6answers
93 views

Simplifying radicals inside radicals: $\sqrt{24+8\sqrt{5}}$

Simplify: $\sqrt{24+8\sqrt{5}}$ I removed the common factor 4 out of the square root to obtain $2\sqrt{6+2\sqrt{5}}$, but the answer key says it is $2+2\sqrt{5}$. Am I missing out on some general rule ...
0
votes
1answer
84 views

What is $\sqrt{a+b}$ in terms of $a$ and $b$?

So a while back I learned that $(a+b)^2 = a^2 + 2ab + b^2$ So you can probably see where that's going, I just want to see what the reverse of that is. What I've tried is this (spoiler alert, it ...
2
votes
1answer
55 views

How can I solve this problem $\sqrt{30 \sqrt{30 \sqrt{30 …}}}$

How can I solve this problem $\underbrace{\sqrt{30 \sqrt{30 \sqrt{30 ...}}}}_{10 \text{ times}}$ Should I calculate it straight forward way adding the indices
1
vote
2answers
38 views

simplification of square root of $\pi$

Using the power rule, my textbook differentiates this: $\frac{d}{dx}(\sqrt{x^{2+\pi}})$ like this, using the power rule: $$\begin{align} =& \frac{d}{dx}(x^{1+(\pi/2)}) \tag{1}\\ =& ...
4
votes
1answer
83 views

Find $\sqrt 7 \pmod {2579}$

Find $\sqrt 7 \pmod {2579}$. I think I understand how I would solve a very basic equation like this: $x^2 = 1 \pmod 5$ make a table of all the possible solutions like this $x=0 \implies x^2=0 \\ ...
3
votes
3answers
563 views

How to calculate this integral with square roots: $\int\frac{ \sqrt{x+1} }{ \sqrt{ x-1 }} \, dx$

How would you calculate this integral: $$\int_{}\frac{ \sqrt{x+1} }{ \sqrt{ x-1 }} \, dx$$
-4
votes
7answers
121 views

Prove that $\frac{1-\sqrt{1-x^2}}{x}\le1$ [closed]

What are different ways to prove that: $$\frac{1-\sqrt{1-x^2}}{x}\le1$$ for $0<x<1$ Thanks!
4
votes
1answer
50 views

Computing square roots with arithmetic-harmonic mean

We know that if we iterate arithmetic and harmonic means of two numbers, we get their geometric mean. So, basically if we need to compute the square root of $x$: $$\sqrt{x}=\sqrt{1 \cdot ...
2
votes
1answer
110 views

How can I prove $\sqrt{\sqrt2}$ to be irrational?

How can I prove $\sqrt{\sqrt2}$ to be irrational? I know that $\sqrt2$ is an irrational number, it can be proved by contradiction, but I'm not sure how to prove that $\sqrt{\sqrt2} = \sqrt[4]{2}$ ...
3
votes
3answers
32 views

Simplification of surds $\frac{x}{\sqrt{x^2 - x^4}}$

$$\frac{x}{\sqrt{x^2 - x^4}}$$ I believe that I can factor out the $x^2$ in the square root to get $$\frac{1}{\sqrt{1-x^2}} .$$ However, Wolfram Alpha doesn't do the simplification, hence my ...
0
votes
1answer
20 views

Defined integrals-Aplications

1) Calculate the area of the function bordered by function graph $f : (0, ∞) → R$, $f (x) = \frac{\sqrt{x}+2}{x+1}$ and axe Ox, lines $x=1$ and $ x =4 $ //On this I get stuck and sqrt(x) under x+1 ...
4
votes
3answers
186 views

Find formula for $\frac{1}{\sqrt 1}+ \frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n}$

I have the series: $$\frac{1}{\sqrt 1}+ \frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n}$$ I find hard to generalize into one formula, any explanation would be helpful.