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

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0
votes
3answers
18 views

Are there any integral solutions to this inequality?

Are there any integral solutions to this inequality? $$\frac{n\sqrt{3} + 1}{n\sqrt{3}} + {\left(\frac{2n}{n + 1}\right)}^{1/2} < 1 + \sqrt{3}$$ WolframAlpha appears to give an inconsistent ...
2
votes
1answer
61 views

Are logarithms radicals?

Does the set of all logarithms with a radical base and argument belong to the set of all radicals? A simple yes, no answer will suffice, an explanation would be wonderful.
3
votes
3answers
217 views

Solving a Radical Equation (squaring doesn't help)

How should I approach this problem: $$ 5(\sqrt{1-x} + \sqrt{1+x}) = 6x + 8\sqrt{1-x^2} $$ I've tried squaring both sides but to get rid of all the radicals requires turning it into a quartic equation, ...
2
votes
1answer
57 views

Solutions of $\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$

$\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$ I get that $-3$ as a solution, but apparently 1 is as well a solution, and I don't see a mechanism through which I could find it. Any help would be ...
3
votes
2answers
58 views

Is it possible to find the sum of all integer values that $x$ can take?

Is it possible to find the sum of all integer values that $x$ can take? In: $$\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$$
5
votes
3answers
94 views

Can $\sin(\pi/25)$ be expressed in radicals

I suspect that sin(pi/25) is not expressible in elementary forms in radicals because it is the root of some quintic (or rather cos(pi/25) is). Can anyone prove that that particular quintic has no ...
1
vote
3answers
48 views

solving $a = \sqrt{b + x} + \sqrt{c + x}$ for $x$

I'm trying to solve a very simple looking square root equation but nothing seems to work. The equation has this form (solve for $x$): $$ a = \sqrt{b + x} + \sqrt{c + x} $$ Squaring both sides ...
-4
votes
1answer
26 views

Question including Function and Radical [closed]

For $f(x)= \sqrt{x-2}+1$ : How would I solve for $x$ make a table and graph? Thank you!
2
votes
2answers
53 views

How would I make a graph for $\sqrt{x+1}-3$

I have to make a table for $\sqrt{x+1}-3$ and I can't figure out how to find the $x$ values. I know that I have to get the middle section to equal perfect squares, which are $0,\ 1,\ 4,\ 9$ but I ...
0
votes
1answer
33 views

Simplifying Radicals (Algebra II Basics) [closed]

Please help me with these problems! I have a quiz coming up and this is just practice. I already solved them, I just am asking for another set of eyes to solve these. (It is not homework, only extra ...
0
votes
1answer
35 views

Recurring Nested Radical of $0$

I'm having difficulty in determining the value of a recurring nested radical where $n = 0$. I am using the equations $x = \dfrac{1 + \sqrt{1 + 4n}}{2}$ and $n = \dfrac{((2x-1)^2 -1)}{4}$ to calculate ...
2
votes
5answers
116 views

Solving $ \sqrt{x - 4} + \sqrt{x - 7} = 1 $.

I have the equation $ \sqrt{x - 4} + \sqrt{x - 7} = 1 $. I tried to square both sides, but then I got a more difficult equation: $$ 2 x - 11 + 2 \sqrt{x^{2} - 11 x - 28} = 1. $$ Can someone tell me ...
43
votes
11answers
7k views

What exactly IS a square root?

It's come to my attention that I don't actually understand what a square root really is (the operation). The only way I know of to take square roots (or nth root, for that matter) it to know the ...
4
votes
3answers
81 views

Calculate fractional part of square root without taking square root

Let's say I have a number $x>0$ and I need to calculate the fractional part of its square root: $$f(x) = \sqrt x-\lfloor\sqrt x\rfloor$$ If I have $\lfloor\sqrt x\rfloor$ available, is there a ...
1
vote
3answers
41 views

If I have the value of $\sqrt{1.3}$ could it be possible to find other square roots from that value? using the manipulation of surds?

If I have the value of $\sqrt{1.3}$ could it be possible to find other square roots from that value? using the manipulation of surds?
0
votes
0answers
70 views

Integration Problem Involving Cube Roots

The question is as follows: $\int\frac{x−2}{(x^2−4)^{1/3}}$ What I tried doing was that I took $x−2=t$. Therefore, $\frac{dt}{dx} = 1$ and $dt = dx $ On simplifying: $\int \frac{x-2}{(x^2 - ...
0
votes
2answers
53 views

If sequence $x_n$ converges to $x$, prove that $\sqrt x_n$ converges to $\sqrt x$ [duplicate]

If sequence $x_n$ converges to $x$, prove that $\sqrt x_n$ converges to c. I know I have to estimate $| \sqrt x_n - \sqrt x |$. But I cannot start. Thanks
3
votes
2answers
63 views

Evalutating $\lim_{x\to +\infty} \sqrt{x^2+4x+1} -x$ [duplicate]

I'm looking to evaluate $$\lim_{x\to +\infty} \sqrt{x^2+4x+1} -x$$ The answer in the book is $2$. How do I simply evaluate this problem? I usually solve limits such as this with the short cut ...
4
votes
1answer
48 views

$\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$

Question : Solve $\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$ My Try: Let u = $1+\frac{9x}{4}$ Then, $$du = \frac{9x}{4}dx$$ $$dx = \frac{4du}{9}$$ Substituting the above in the main ...
0
votes
2answers
40 views

How to find the limit of a function with a third root?

I have the function $x^2(\sqrt[3]{x^3+1} - x) $ and have to find the limit for $x \rightarrow \infty $. After many hours of forming around I still have no clue how to find it. Is there anybody who ...
5
votes
4answers
128 views

If $a,b$ are positive rational numbers and $\sqrt a+\sqrt b$ is rational, then both of $\sqrt a,\sqrt b$ are rational numbers

I'm trying to show that If $a,b$ are positive rational numbers and $\sqrt a+\sqrt b$ is rational, then both of $\sqrt a,\sqrt b$ are rational numbers. I squared the number $\sqrt a+\sqrt b$ ...
0
votes
3answers
47 views

Square Root Confusion

well we know that $$\sqrt{x^2} = \pm x$$ Then if $$x^2=y^2$$ then $$\pm x= \pm y$$ Does this mean $x = y$ or $-x = -y$ or $x = -y$ or $-x = y$ or all is true? Which is true among these?
45
votes
3answers
714 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 ...
1
vote
2answers
50 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
54 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 = ...
19
votes
6answers
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
28 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
52 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
121 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
33 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
80 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 ...
5
votes
4answers
582 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
36 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
103 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
44 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, ...
1
vote
2answers
55 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) = ...
0
votes
0answers
27 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
115 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
57 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 - ...
25
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
85 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
191 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
99 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
123 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
55 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
142 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
117 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}}}} + ...