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Questions tagged [radicals]

For questions involving radical of numbers or radical of expressions (i.e. numbers/expressions raised to the power of a fraction).

-4
votes
2answers
30 views

How to find integer solution of an equation [on hold]

How to find all integer solution that satisfies the equation: $$4x+y+4\sqrt{xy}-28\sqrt{x}-14\sqrt{y}+48=0$$
6
votes
2answers
55 views

Sum of 5 square roots equals another square root. What is the minimum possible value of the summed square root?

In the equation $\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}+\sqrt{e}=\sqrt{f}$, each variable is a distinct positive integer. What is the least possible value for $f?$ Out of purely trial and error, I have ...
3
votes
2answers
31 views

equation with exponential functions 2

Solve the following equation over the real numbers: $$ (3+ \sqrt{5})^x + (3- \sqrt{5})^x=7 * 2^x $$
0
votes
2answers
72 views

What is the best method to square root a number by head?

I watched some movies on some people do square roots by head, Like in Rainman and The Man Who Knew Infinity. Did they just know by heart the roots or did they have some easy method to do it on head?
1
vote
3answers
58 views

Evaluate the values of $x$ in $\sqrt{2x-5} = \sqrt[3]{6x-15}$

$$\sqrt{2x-5} = \sqrt[3]{6x-15}$$ Evaluate the values of $x$ I believe that there would be an easier approach to this problem because I will have to expand 3th degree binomial as shown ...
1
vote
1answer
26 views

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $ I feel like the products between different variables (i.e. not x^2, y^2, z^2) give this inequality the $ \gt $ sign and I don't ...
2
votes
3answers
59 views

Calculate limit with squares $\lim_{n \to \infty}\left(\sqrt[3]{n^3+2n-1}-\sqrt[3]{n^3+2n-3}\right)^6 \cdot (1+3n+2n^3)^4$

$$\lim_{n \to \infty}\left(\sqrt[3]{n^3+2n-1}-\sqrt[3]{n^3+2n-3}\right)^6 \cdot (1+3n+2n^3)^4 $$ What I did was to multiply it and I got $\frac{1}{2}$ as the final result. Could someone confirm if it'...
1
vote
1answer
40 views

Some interesting systems of equations [closed]

1, Solve the system of equations:$\left\{\begin{matrix} x^3+y^3+2z^3=19x-11y-5z+1\\ x^3+(y^2+1)x=(x^2+y^2)z+z \\ \sqrt{2+x^2+y^2-2yz}=y^2+z^2-2xy+\sqrt{2} \end{matrix}\right.$ 2,Solve the system of ...
2
votes
1answer
49 views

Explain the process of solving this nested squareroot problem [closed]

I am in class 10 and this is an olympiad question so I am having a problem doing this. The innermost one I have evaluated to be $\sqrt{15}- \sqrt2$. But the rest I don't know how to do. $\sqrt { 2 + \...
0
votes
0answers
40 views

Proof that 2*2/sqrt(2)*2/sqrt(2+sqrt(2))*2/sqrt(2+sqrt(2+sqrt(2)))*… equals PI?

I found this formula that $\Pi=2*2/\sqrt{2}*2/\sqrt{2+\sqrt{2}}*2/\sqrt{2+\sqrt{2+\sqrt{2}}}*...$ I tested it out and it seems to be true, but I don't get why this is. Searching for this only lead ...
0
votes
2answers
60 views

How can I simplify $\sqrt{\frac{5+\sqrt{5}}{2}}$?

I've tried to see the root as $\sqrt{\frac{5+\sqrt{5}}{2}} = \sqrt{a}+\sqrt{b},$ but this method doesn't give me something good.
1
vote
3answers
83 views

How do I simplify $\sqrt {4(2- \sqrt{3})}$ into $\sqrt{6} - \sqrt{2}$

This might be a stupid question, but how do I get from $$\sqrt {4(2- \sqrt{3})}$$ to $$\sqrt{6} - \sqrt{2}$$ It is obvious if you squared both, they both equal $8 - 4 \sqrt{3}$, but I'm wondering how ...
0
votes
2answers
79 views

Proof Verification - Elementary proof that $\sqrt3$ is irrational

Sorry for the dumb question; something about this proof seems off and I was wondering what (if anything) is wrong with it. Assume $p$, $q$ are integers. We prove by contradiction. $\sqrt3 = p/q$ $(...
0
votes
1answer
24 views

Finding the rationalizing factor of rational numbers with denominator 1

I have a question which I could not solve after hours of research. It goes like this: Find the rationalizing factor of $$\sqrt[3]{16} - \sqrt[3]{4} + 1$$ I can rationalize the denominator but can’...
3
votes
1answer
72 views

If positive $a$, $b$, $c$, $d$ satisfy ${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$, then $abcd\geq 81$

Let $a,b,c,d>0$ satisfying $${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$$ Prove that $abcd\geq 81$ I've tried to apply arithmetic geometric mean inequality or Cauchy-Schwartz ...
2
votes
2answers
44 views

How does $d_1$ equal $\frac{1}{2}\sqrt{\left(x_2-x_1\right)^2 + \left(y_2-y_1\right)^2}$?

I'm going over the proof of the midpoint formula and the solution in my textbook solves its first distance as follows $$d_1 = \sqrt{\left(\frac{x_1+x_2}{2}-x_1\right)^2 + \left(\frac{y_1+y_2}{2}-y_1\...
3
votes
0answers
57 views

Additional values of Dedekind's $\eta$ function in radical form

Can anyone confirm the following values of the $\eta $ function to increase the table of the post What is the exact value of $\eta(6i)$? ? $\eta(9i)$ = $\frac{1} {6} \big(\sqrt{6}\, (2+\sqrt{3})^{1/...
0
votes
3answers
41 views

Finding rational numbers in an equation with two variables

How should we find two rational numbers $\alpha$, $\beta$ such that $\sqrt[3]{7+5\sqrt{2}}=\alpha+\beta\sqrt{2}$? The answer I got alpha = 1 and betta = 1. If I'm wrong, please correct me. Thank you
1
vote
3answers
67 views

If $3 - 5i$ is a square root of $z$, find the other root.

If $3 - 5i$ is a square root of $z$, find the other root. Well, I was under the impression that only the sign in front of the imaginary part would change so the other root would be $3 + 5i$. However,...
4
votes
3answers
230 views

$\sqrt{24ab+25}+\sqrt{24bc+25}+\sqrt{24ca+25}\geq 21$ if $a+b+c=ab+bc+ca$?

For $a,b,c>0 $ and $a+b+c=ab+bc+ca$ . Prove or disprove that : $\sqrt{24ab+25}+\sqrt{24bc+25}+\sqrt{24ca+25}\geq 21$ I checked in very many cases. Example :$c=1, a=2,b=\frac{1}{2}...$ then it’s ...
2
votes
1answer
33 views

Let $[L:K]=2$ and char $K\neq 2$ show that $L/K$ is a simple 2-radical extension.

I want to show that $[L:K]=2$ and char $K\neq 2$ $\Rightarrow$ $L/K$ is a simple 2-radical extension. I know that, since $[L:K]$ is prime, the extension is simple. I also concluded that, since $[L:K]...
20
votes
3answers
3k views

Why is Sesame Street's Count von Count's favorite number $34,\!969$? [closed]

In the 2 minute BBC News audio clip Sesame Street: What is Count von Count's favourite number? "The Count" is asked Do you have a favorite number? to which he replied Thirty four thousand, nine ...
0
votes
1answer
59 views

Find anti-derivative of hairy expression under radical

I'm lost on how to pull the anti-derivative of the expression under the radical. The full integral I'm evaluating is: $$2\pi\int_{16}^{25}(9-{\sqrt{x}})^2{\sqrt{1+\left(\frac{9}{\sqrt{x}}\right)^2}}...
0
votes
0answers
21 views

Find the intersection points between a pyramid and 4 lines

I'm trying to implement a real time location tracking algorithm but I have some issues resolving the equations I have. The idea is to find the coordinates of the corners of a triangular pyramid with ...
1
vote
1answer
15 views

Semiprimitive rings and Modules

I'm finding for simple proofs of the followings facts: $\bullet$ rad$(\Lambda/rad \Lambda)=0$ $\bullet$ rad$(A/rad \Lambda \ . A)=0$ $\bullet$ rad$(A/rad \Lambda \ . A)= rad A/ (rad\Lambda. A)$ ...
3
votes
0answers
48 views

Integral of $\sqrt{a(b(x+c)^2+1)}/((x-1)x^{3/2})$

I am trying to solve $$\int_{x_0}^\infty\frac{\sqrt{a(b(x+c)^2+1)}}{(x-1)x^{3/2}}dx$$ where, $a,b,c,x_0,x\in\Bbb R$ and $b,c,x>0$ and $x_0>1$. I tried a $\sinh$ substitution without too much ...
2
votes
2answers
122 views

Tough Irrational Equation highschool

Have been trying to solve this irrational equation for a day but as it seems, i'm not going anywere with it. Can somebody offer me a tip ? Thanks! *Tried a "t" substitution for x squared but it still ...
3
votes
1answer
43 views

Square roots of $j$ and $ε$

I know how to find the square root of the imaginary unit $i$, but I'm still learning about split-complex and dual numbers. I can't find any info anywhere about the square roots of $j$ and $ε$, if they ...
2
votes
1answer
68 views

Is it guaranteed that $\sqrt[3]{a+\sqrt{b}}$ can be denested with or without complex numbers?

I tried to use the cubic formula before, but was always stuck at simplifying the cube root. I had learned that you can always simplify $\sqrt{a+\sqrt{b}}$ by solving, but can I always simplify $\sqrt[...
-1
votes
3answers
41 views

Solve cubic root equation [closed]

$ \sqrt[3]{x+5} - \sqrt[3]{x-5} = 1 $ How many values of x satisfy the equation?
2
votes
3answers
72 views

Problem when convert $\sqrt{A}+\sqrt{B}=\sqrt{C}+\sqrt{D}$ to A+B=C+D.

This is what my lecturer taught me. If you have $\sqrt{A}+\sqrt{B}=\sqrt{C}+\sqrt{D}$ You can easily convert to $A+B=C+D$ or $AB=CD$ And then he gave me an example. $\sqrt{8x+1}+\sqrt{3x-5}=\sqrt{...
2
votes
1answer
97 views

Where to start with: $\lim_{n\to\infty} (\sqrt[3]{n^{48}+n} - \sqrt[3]{n^{48}+n^2}) ((n^3 +3)^{12} - (n^4+4n)^9)$

I have limit: $\lim_{n\to\infty} (\sqrt[3]{n^{48}+n} - \sqrt[3]{n^{48}+n^2}) ((n^3 +3)^{12} - (n^4+4n)^9)$ I have to find that it is equal to -6 but I do not know how. What I did was to get rid of ...
0
votes
2answers
61 views

Attempting to prove the following inequality

I got a question on a test to prove the following inequality: $$ \sqrt{1*2}\space+\space\sqrt{2*3}\space+\space...\space+\space\sqrt{n(n-1)}>\frac{n^2-1}{2} : n>1, n\in\mathbb{N} $$ I tried to ...
0
votes
1answer
59 views

How to find minimum value

For $a=\sqrt{x^2-3\sqrt2x+9}$ and $b=\sqrt{x^2-5\sqrt2x+25}$ what is the value of $x$ when $a+b$ is minimum and how to find this? Thanks in advance.
7
votes
3answers
1k views

Proving irrationality of $\sqrt[3]{3}+\sqrt[3]{9}$ [duplicate]

I need to prove $$\sqrt[3]{3}+\sqrt[3]{9}$$ is irrational, I assumed $$\sqrt[3]{3}+\sqrt[3]{9} = \frac{m}{n}$$ I cubed both sides and got $$\sqrt[3]{3}+\sqrt[3]{9} = \frac{m^3-12n^2}{9n^3}$$ I tried ...
-1
votes
2answers
54 views

Find min of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+4\sqrt{2}\sqrt{\frac{ab+bc+ac}{a^2+b^2+c^2}}$ [closed]

Given the three real numbers a, b, c are not negative, in which at most some are equal to zero. Find min of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+4\sqrt{2}\sqrt{\frac{ab+bc+ac}{a^2+b^2+c^2}}$ ...
2
votes
3answers
140 views

What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}, x \in \mathbb{R}$ if $\sqrt{49-x^2}-\sqrt{25-x^2}=3$

Suppose that real number $x$ satisfies $$\sqrt{49-x^2}-\sqrt{25-x^2}=3$$What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$? This is what I did: I try to multiply by the conjugate. Its value I believe ...
0
votes
1answer
25 views

Square-root equation with no solutions

There is an equation given: $${\sqrt {3x-7}} + {\sqrt {2x-1}} = 0$$ Solving it algebraically: $${(\sqrt {3x-7})^2} + {(\sqrt {2x-1})^2} = 0$$ $$ 3x-7 + 2x-1 = 0 $$ $$ 5x = 8 $$ $$ x = \frac{8}{5} $$...
2
votes
1answer
39 views

Radical of $I = (xy-x^3, y^2-yx^2)$

Quick question about finding $\sqrt I$, for $I = (xy-x^3, y^2-yx^2) \subset k[x,y]$. Attempt: Using for $I, J$ ideals we have $\sqrt{IJ} = \sqrt I \cap \sqrt J$. $\sqrt{(xy-x^3, y^2-yx^2)} = \sqrt{...
1
vote
1answer
56 views

Please explain why 2x * /2 became 2/2x

I'm studying a Grade 10 maths book and was surprised to see the correct answer for this question was: (/2 - 3x)(/3 + 2x) - /5x = /6 + 2/2x - 3/3x - 6x - /5x I thought it should be: = /6 + 2x/2 - ...
1
vote
2answers
56 views

Simplifying Radicals In Heron’s Formula

When I sometimes use Heron’s formula and the Pythagorean Theorem in the coordinate system to find the area of a triangle, I get stumped at the last step: simplifying the radical. Is there a general ...
1
vote
1answer
48 views

Different results with different methods to same limit of polynomial

I'm considering the following limit while looking for asymptotes: $$\lim_{x\to \infty} \arctan\dfrac{\sqrt{x^2+1}}{x-1}$$ Going the route I initially tried, I divide both parts of the fraction by $x^...
0
votes
1answer
41 views

While solving an equation I get 2 answers, but when I substitute one of the answers the equality doesn't hold?

I solved the following equation: $$\sqrt{x + 1} + \sqrt{2 \cdot x + 3} - \sqrt{8 \cdot x + 1} = 0$$ I get 2 answers $3,-1/17$ but when I plug $-1/17$ on the equation the equality is wrong.
3
votes
3answers
76 views

If x,y and z are positive integers and $\frac 1x + \frac 1y = \frac 1z$ then $\sqrt{x^2+y^2+z^2}$ is rational.

To solve this problem I first started off by factoring to get $z^2=(x-z)(y-z)$ only to realise that this does nothing so I then tried squaring both sides to get the reciprocals of $x,y$ and $z$ ...
1
vote
3answers
74 views

prove the inequality $\left({1-x\over1+x}\right)^{1/2}\lt{\ln(1+x)\over \arcsin(x)}\lt1$ for $x\in (0,1)$

$$\left({1-x\over1+x}\right)^{1/2}\lt{\ln(1+x)\over \arcsin(x)}\lt1$$ for $x\in(0,1)$ My attempt: For the upper bound, I took the derivative of ${\ln(1+x)\over \arcsin(x)}$ and found out its a ...
6
votes
1answer
138 views

Inequality. ${{\sqrt{a}+\sqrt{b}+\sqrt{c}} \over {2}} \ge {{1} \over {\sqrt{a}}} + {{1} \over {\sqrt{b}}} + {{1} \over {\sqrt{c}}}$

Question. If ${{a} \over {1+a}}+{{b} \over {1+b}}+{{c} \over {1+c}}=2$ and $a$, $b$, $c$ are all positive real numbers, prove that $${{\sqrt{a}+\sqrt{b}+\sqrt{c}} \over {2}} \ge {{1} \over {\sqrt{a}}}...
0
votes
0answers
37 views

Algorithm for regular continued fraction of a square root

Say I have a number $n$, and want to find the expression of $\sqrt{n}$ as a regular continued fraction. How would I do such a thing systematically? A naive computer algorithm wouldn't work, due to ...
1
vote
1answer
57 views

Simplifying a solution to $x^2 = 4 + 2\sqrt{2}$

$$x^2 = 4 + 2\sqrt{2}$$ $$x = 2 + \sqrt{2\sqrt{2}}$$ Neglecting that the result can be negative as well, how should I continue? How can I simplify it?
4
votes
2answers
70 views

Different ways to express $\sqrt{a}+\sqrt{b}$

I had been thinking about this for a long time. I can’t express $\sqrt{a}+\sqrt{b}$ in ways that are useful. (I’m not wanting a formula, but just some ways to express this.) I can only think of $\sqrt{...
0
votes
0answers
39 views

What method of finding the square root they use in this screenshot?

I saw somewhere a screenshot from a old(?) Russian cartoon with the following math inside. On a screenshot one can see that a student successfully solves 2 exercises on finding square root and ...