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

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12
votes
0answers
61 views

Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$

When I was playing with numbers, I found that there are many triplets of three positive integers $(a,b,c)$ such that $\color{red}{2\le} a\le b\le c$ $\sqrt{abc}\in\mathbb N$ $\sqrt{abc}$ divides ...
0
votes
2answers
38 views

Simplifying Square Roots Frustration

Okay, I'm really frustrated with this. So, when you have $3 \sqrt 5 + 5 \sqrt 5$, you get $8\sqrt5$, right? Okay, so what do I do for here: $\sqrt{11} - 3 \sqrt{11}$ Is it just $-3 \sqrt{11}$ ? ...
2
votes
0answers
30 views

Sum of Floor of Square Root: $S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor$

$$ S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor. $$ Hello, I´m trying to solve this summation. I was able to get $$ a_n = 2a_{(n-1)} - a_{(n-2)} $$ for non perfect square numbers and $$ a_n = ...
18
votes
7answers
2k views

Why aren't these negative numbers solutions for radical equations?

I was working on radical equations and I came across a few problems where I got answers that worked when I checked, but were not listed as solutions. My teacher's only explanation was, "just because." ...
1
vote
1answer
25 views

Is the nth root of a product of n terms used in place of the average anywhere?

In applied usage we typically take the average of values or terms which is done by summing them and dividing by the number of terms (for simple average): $$\sum_{i=1}^n \frac{a_i}{n}$$ It dawned on ...
0
votes
3answers
26 views

Complex values of the cube root

I just learned that the cube root has 2 complex roots. For example, the cube root of 8 has : 2 , -1 plus or minus square root of 3 *i I was wondering, how do you find those conjugate complex values ...
1
vote
3answers
41 views

System of equations with radicals

Solve the system of equations (in $\mathbb R$): $$\begin{matrix} 2\sqrt[4]{\frac{x^4}{3}+4}=1+\sqrt{\frac{3}{2}y^2} \\ 2\sqrt[4]{\frac{y^4}{3}+4} = 1+\sqrt{\frac{3}{2}x^2} \end{matrix}.$$ This ...
10
votes
1answer
104 views

Identity in Ramanujan style

Is it possible to represent $$ \sqrt[3] {7\sqrt[3]{20}-1} =\sqrt[3]{A}+\sqrt[3]{B}+\sqrt[3]{C}$$ with rational $A,\,B,$ and $C?$
1
vote
2answers
30 views

Why is the Jacobson radical of the integers {0}.

Why is the Jacobson radical of the integers {0}? I have been working through questions dealing with the Jacobson radical and have come across this and can't think of why this would be. Any help ...
2
votes
3answers
71 views

Prove irrationality of $\sqrt{2+\sqrt{2}}$ and $\sqrt{2}+\sqrt{3}$ [duplicate]

I am trying to prove the irrationality of the above two numbers, but I don't know how. What would be a general strategy for problems like these? My current strategy is trying to reach a contradiction ...
0
votes
1answer
20 views

Vertical asymptotes of a given non-rational radical funtion

We have that $f$ is a function $f(x) = x\sqrt{x+4}$. Hence, $f'(x) = \dfrac{3x+8}{2\sqrt{x+4}}$. Then, $\lim_{x \to -4^+}f'(x) = -\infty$. This means that $f$ has a vertical slope at $f(-4)$. It ...
4
votes
4answers
554 views

Can this be shown: $\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\dots}}} = \sqrt a$?

$$\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\sqrt[3]{a\cdots}}}}}}}}}=\sqrt{a}$$ Just for fun. I would like to read the proof of this if it exists. Any ...
-1
votes
1answer
33 views

Simple division and some radicals

I'm having a lot of trouble figuring out how to evaluate this problem: $$\dfrac{7}{\frac{7\sqrt{85}}{85}}$$ Everything I've found says that the answer is just $1\sqrt{85}$, but I have no idea how to ...
3
votes
2answers
97 views

Prove $ \sum \frac{\cos n} { \sqrt n}$ converges

How to Prove $ \sum \frac{\cos n} { \sqrt n}$ converges Using Abel’s theorem ? I think it can be done using $\cos n = Re(e^{in})$ { Real Part of Complex Number } How to proceed ?
1
vote
2answers
43 views

Why does this approximation of square roots using derivatives work this way?

I came up with this way to estimate square roots by hand, but part of it doesn't seem to make sense. Consider how $f(n) = \sqrt{n^2+\varepsilon} \approx n$ when $\varepsilon$ is small. Therefore, ...
0
votes
2answers
41 views

Two definitions of Jacobson Radical

I have in my notes that the Jacobson radical of a ring $R$ is: $J(R) = \cap${$I$ | $I$ primitive ideal of $R$} $= \cap$ {$Ann_R M$ | $M$ simple $R$-module}. I have now seen elsewhere that J(R) = {x ...
0
votes
0answers
26 views

closed form or approximate solution for a system of equation : $m(t)=v\sin(\arctan(at+b))+v\sin(\arctan(ct+d))$

Can one solve for $(v,a,b,c,d)$ the following equation ? $t$ takes discrete values and $m(t)$ is known for as many $t$ needed. However please assume that special values of $t$ may not be available ...
1
vote
3answers
110 views

How to express $\sqrt{x} =-1$?

How would one express a solution to $\sqrt{x} =-1$? I just read that a solution to the above equation cannot be expressed in the form of complex numbers, really interested in any additional ...
6
votes
3answers
49 views

Simplify $\left(\sqrt{\left(\sqrt{2} - \frac{3}{2}\right)^2} - \sqrt[3]{\left(1 - \sqrt{2}\right)^3}\right)^2$

I was trying to solve this square root problem, but I seem not to understand some basics. Here is the problem. $$\Bigg(\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\bigg)^2} - \sqrt[3]{\bigg(1 - ...
24
votes
6answers
1k views

How to prove that $\sqrt[3] 2 + \sqrt[3] 4$ is irrational?

So while doing all sorts of proving and disproving statements regarding irrational numbers, I ran into this one and it quite stumped me: Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational. I tried ...
1
vote
3answers
74 views

Average of square roots's sum vs. square root of an average

I was watching a video on youtube about how colors work in computers, and found this statement: "The average of two square roots is less than the square root of an average" The link to the ...
0
votes
0answers
23 views

How to formally show a field extension is not radical

I'm wondering if there is a general procedure for showing that a field extension is not radical. As an example, let $L=\mathbb{Q}(\sqrt[3]{1+\sqrt{2}})$. Then I can see that $L/\mathbb{Q}$ isn't a ...
2
votes
6answers
189 views

A quick way to prove the inequality $\frac{\sqrt x+\sqrt y}{2}\le \sqrt{\frac{x+y}{2}}$

Can anyone suggest a quick way to prove this inequality? $$\frac{\sqrt x+\sqrt y}{2}\le \sqrt{\frac{x+y}{2}}$$
5
votes
2answers
94 views

Find conditions on positive integers so that $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is irrational

Find conditions on positive integers $a, b, c$ so that $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is irrational. My solution: if $ab$ is not the square of an integer, then the expression is irrational. I ...
0
votes
3answers
122 views

Prove that $3 - 2 ^ {1/7}$ is Irrational

How to prove that $3 - 2 ^ {1/7}$ is irrational? If I do $$\frac p q = 3 - 2 ^ {1/7}$$ $$2 ^ {1/7} = 3 - \frac p q $$ Hint needed Should I multiply by $7$ times??
0
votes
3answers
52 views

Solving a radical equation for real roots

I'm attempting to solve the derivative of my function $f(x)$ for real roots. $$ \\ \begin{align*} \\ f(x) &= 3x^2 + 3\arcsin{x} \\ f^{\prime}(x) &= 6x + \dfrac{3}{\sqrt{1-x^2}} \\ \\ 0 &= ...
2
votes
6answers
130 views

What type of number is this $\frac{\sqrt2}{2}$?

$$\frac{\sqrt{2}}{2}$$ In this monomial, an irrational number is divided by a rational number. However this is not a general case but can any one tell me that when we divide an irrational number or ...
4
votes
1answer
114 views

Simplifying $\scriptsize\sqrt{2+\sqrt{2}} + \sqrt{2+\sqrt{2+\sqrt{2}}} + \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}} + \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}$

The question is in the title: is there a simpler form or result for $$\sqrt{2+\sqrt{2}} + \sqrt{2+\sqrt{2+\sqrt{2}}} + \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}} + ...
8
votes
5answers
213 views

Prove that $\sqrt{3}+ \sqrt{5}+ \sqrt{7}$ is irrational

How can I prove that $\sqrt{3}+ \sqrt{5}+ \sqrt{7}$ is irrational? I know that $\sqrt{3}, \sqrt{5}$ and $\sqrt{7}$ are all irrational and that $\sqrt{3}+\sqrt{5}$, $\sqrt{3}+\sqrt{7}$, ...
0
votes
1answer
31 views

Galois group and solvable by radicals

I came across the following problem in an old qualifying exam which states: Show that the irreducible $h(x)\in \mathbb{Q}[x]$ is solvable by radicals if $[K:\mathbb{Q}]=25$ where $K$ is the ...
3
votes
1answer
45 views

Which mixed numbers have the property $\sqrt{n + \frac{p}{q}}=n\sqrt{\frac{p}{q}}$?

Could I please have help with describing mixed numbers (aka mixed fractions) that have this property: Show that $\sqrt{9\frac{9}{80}}=9\sqrt{\frac{9}{80}}$ and ...
1
vote
3answers
59 views

Expressing $\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$ with rational Denominator

could you please help me express this with a rational denominator $\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$ Thank you
2
votes
2answers
68 views

Rationalizing the fraction $\frac{1}{1-\sqrt2 -\sqrt3}$

I'm having problem in rationalizing the following root with the fraction $$\frac{1}{1-\sqrt2 -\sqrt3}$$ Eventually after many tries, I found the solution which was : $$\frac{-\sqrt2 (1-\sqrt2 +\sqrt3 ...
4
votes
4answers
178 views

How to compute a lot of digits of $\sqrt{2}$ manually and quickly?

After having read the answers to calculating $\pi$ manually, I realised that the two fast methods (Ramanujan and Gauss–Legendre) used $\sqrt{2}$. So, I wondered how to calculate $\sqrt{2}$ manually in ...
1
vote
2answers
51 views

Difference of Square root and order relation

If $a=\sqrt{15} -\sqrt{11}$, $b=\sqrt{27} - \sqrt{23}$, $c=\sqrt{6} - \sqrt{5}$ then which of the following is correct? ...
2
votes
3answers
67 views

How to simplify $(a^2+ab+b^2)/(a+\sqrt{ab}+b)$

How can I simplify as much as possible: $$\frac{a^2+ab+b^2}{a+\sqrt{ab}+b}$$ Also, first post here, looking forward to sticking around!
3
votes
3answers
78 views

How to write $1/ \left(1+\sqrt 3+\sqrt 5+\sqrt{15}\right)$ with a rational denominator?

How to write $\frac{1}{1+\sqrt{3}+\sqrt{5}+\sqrt{15}}$ with a rational denominator? There is an included hint: factorize the denimator Edit: There has been some confusion on this question, the first ...
0
votes
4answers
154 views

How to simplify $\sqrt{4-2\sqrt{3}}-\sqrt{3}$ [closed]

I have faced an expression that I cannot simplify. $$ \sqrt{4-2\sqrt{3}}-\sqrt{3} $$ Hope to you will be able to help me with this simple exercise :)
4
votes
1answer
57 views

$f(x) = k^n$ for infinitely many integers $k$

Let $f(x)$ be a polynomial of $n^{th}$ degree with integer coefficients and let the leading coefficient be 1. Is it true that $f(x) = k^n$ for infinitely many integers $k$ and $x$ if and only if all ...
7
votes
1answer
76 views

Can the general septic be solved by infinitely nested radicals?

I. Quintic. The general quintic can be reduced to the form, $$x^5=p+x\tag1$$ $$x = \sqrt[5]{p+x}$$ Hence by an iterative process, $$x =\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+x\dots}}}}$$ ...
6
votes
1answer
62 views

Taking the cube root of a sum of radicals

I am wondering how to derive the following simplification without knowing it beforehand: $$^3\sqrt{10 + 6\sqrt{3}} = 1 + \sqrt{3}$$ After the fact, it is easy to verify algebraically. The problem ...
1
vote
1answer
49 views

Proving an inclusion related to algebraic sets and interpreting it

I want to prove that $I(X_1 \cap X_2) = \sqrt{I(X_1)+I(X_2)}$ for algebraic sets $X_1=Z(G_1)$ and $X_2=Z(G_2)$, with $G_1,G_2 \subseteq \mathbb{K}[X_1,\ldots,X_n]$. Remark: Unfortunately I ...
1
vote
0answers
50 views

Infinitely nested radicals

In a recent paper is is stated ( and maybe proved) that we can solve any polynomial equation with nested radicals. Here "nested radicals" means expression such as: $$ ...
3
votes
2answers
64 views

If x and y are both greater than or equal to 1, show that $|\sqrt{x}-\sqrt{y}|$ is less than or equal to $0.5| x-y |$

If x and y are both greater than or equal to 1, show that $|\sqrt{x}-\sqrt{y}|$ is less than or equal to $0.5| x-y |$ Would really appreciate any help! Thanks
2
votes
5answers
70 views

Is there a way, in general, to tell whether the nth root of a integer is rational?

Is there a way, in general, to tell whether the $n^{th}$ root of a integer is rational? More explicitly, is it possible to elegantly determine whether the result of $k^{1/n}$ is rational for $k,n \in ...
1
vote
2answers
18 views

Roots in combination with exponents

I'm looking for a tip on how to simplify this problem.
0
votes
1answer
35 views

Want to show that $g\in I$ where $I$ is an ideal, given the following conditions

Let $R=K[x_1,...,x_n]$ and $I$ be an ideal of $R$, $K$ being a field Given $h\in I$, $g\in \sqrt{I}$ and $f\in\sqrt{I}$ Where $in_<(f)=in_<(h)$ and $g=f-h$. So $in_<(g) < ...
0
votes
1answer
33 views

Calculating expressions to the power of two with radicals

How do we calculate $(2\sqrt{22})^2$? I tried but failed: $$ 2*2+2*\sqrt{22}+ \sqrt{22}*2+\sqrt{22}*\sqrt{22} $$ The answer is 88. Thanks!
1
vote
1answer
99 views

Is $\sqrt{x^2} = x$?

Does the $\sqrt{x^2}$ always $= x$. I am trying to prove that $i^2 = -1$, but to do that I need to know that the $\sqrt{(-1)^2} = -1$. If that is true then all real numbers are imaginary, because an ...
6
votes
5answers
1k views

What happens when square root is performed in inequalities?

Simplify: $x^2 > 1$. My solution: Taking square root on both sides: $±x > ±1$ So my results are: $x > 1$ $x > -1$ $-x > 1$ $\implies$ $(-1 > x)$ $-x > -1$ $\implies$ $(1 > ...