Questions tagged [radicals]
For questions involving radical of numbers or radical of expressions (i.e. numbers/expressions raised to the power of a fraction).
3,176
questions
-2
votes
1answer
59 views
What is the best way to do X^2= 618
What would be the best and easiest way to do X^2 = 618 (Without a calculator)
Wouldn't it just be considered irrational
(Solving for x)
0
votes
1answer
21 views
Ratio of two sums with inverse radicals
Saw this challenge problem:
$$\frac{\sum_{n=0}^{n=\infty} \frac{1}{\sqrt{3n+1}} - \frac{1}{\sqrt{3n+2}}}{\sum_{n=0}^{n=\infty} \frac{1}{\sqrt{6n+1}} - \frac{1}{\sqrt{6n+5}}} = 2 - \sqrt2$$
How to ...
3
votes
4answers
163 views
Proving that $\sqrt[3] 7 -\sqrt 2$ is irrational.
I understand proving that $\sqrt{7}-\sqrt {2}$ is irrational, but how does the answer change if its cube root of $7$ instead of square root?
the way I solve $\sqrt{7}-\sqrt {2}$ is by assuming its ...
2
votes
1answer
54 views
If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.
If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.
My try
It is easy to see that if we raise the first equation ...
3
votes
1answer
18 views
What is the proper convention regarding the order of operations of a fractional exponent and/or the simplification of it?
Specifically, consider the example $\sqrt[4]{x^2}$. The answer of course would be $\sqrt{|x|}$ since the x is squared first. However if converted to the exponential fraction of $x^{2/4}$, you lose the ...
0
votes
0answers
53 views
Why does this nice method work for expressing accurate trigonometric values in the form $\sqrt{\frac{2\pm\sqrt{2\pm\sqrt{2\cdots\pm\sqrt{2}}}}{2}}$?
I am amazed by the nice work of Mr. Daahal on Breaking-Classical-Rules-in-Trigonometry-Exact-Trigonometric-Values. He provided the algorithm without proof. If someone can provide insight on why ...
1
vote
0answers
37 views
How to find out if sum of square roots is a perfect(-ish) square: $\sum_{i=0}^m \sqrt{x_i} = \bigl(\sum_{j=0}^n \sqrt{y_j}\bigr)^2$?
I'm writing a Python library that deals with symbolic computations of square roots (since approximated ones cause a lot of problems and at least for now correctness is more valuable than performance).
...
2
votes
2answers
43 views
Converting a radical to a fractional exponent
I want to understand how to convert a radical to a fractional exponent. Given the following equation:
$\sqrt[3]{(x)^6\cdot x^9}=\sqrt[3]{x^{24}\cdot x^9}=\sqrt[3]{x^{33}}=x^{\frac{33}3}=x^{11}$
How ...
-2
votes
5answers
61 views
Solving $\lim_{x\to{a}}\frac{x^2-\sqrt{a^3x}}{\sqrt{ax}-a}$ without L'Hopital/derivates [closed]
$$\lim_{x\to{a}}\frac{x^2-\sqrt{a^3x}}{\sqrt{ax}-a}$$
It should be $3a$, but I can't find the way to solve it without L'Hopital.
4
votes
0answers
41 views
Prove that solvability by radicals does not depend on the choice of splitting field
Exercise of Rotmann Abstract Algebra:
If $E/k$ and $E′/k$ are splitting fields of $f(x) \in k[x]$ and there is a radical extension $K_t/k$ with $E ⊆ K_t$, prove that there is a radical extension $K_{r′...
0
votes
1answer
34 views
Square root of irrationals
The Art of Problem Solving: Volume 1 by Sandor Lehoczky and Richard Rusczyk - Example 6-14
We are trying to solve the system $xy = -12$ and $x^2 + 2y^2 = 34$, which will eventually help us solve for ...
3
votes
2answers
418 views
Is the square root of $5^n -1$ an integer only when $n = 1$? [closed]
When is $$\sqrt{5^n-1}$$ an integer if assume that n is a natural number?
It is trivial that it as an integer if $n = 1$ and it can't be an integer if $n$ is even.
-2
votes
5answers
94 views
Determine the sign of $169\cdot7-49\sqrt{123}$
Determine the sign of the discriminant of the equation $$49x^2-26\sqrt7x+\sqrt{123}=0.$$
The coefficients $a,b$ and $c$ of the equation are: $$a=49,\\b=-26\sqrt7\Rightarrow k=-13\sqrt7,\\c=\sqrt{123}.$...
0
votes
1answer
51 views
Addition inside root problem [closed]
Help me calculate the answer for$$ \sqrt{3+\sqrt 5}\cdot \sqrt{3-\sqrt 5}$$ (NO CALCULATOR ALLOWED).
The answer in the book is "2" but I don't know how they solved it.
2
votes
1answer
24 views
Optimal way to pick the starting $x$ when computing the square root with the Babylonian method
I was reading the example given in the Wikipedia article for the Babylonian method of computing the square root, and I wondered why did they set the starting $x_0$ to 600:
To calculate $\sqrt S$, ...
1
vote
2answers
53 views
Help with inverse proportions
The value of $y$ varies inversely as $\sqrt x$ and when $x=24$, $y=15$.
What is $x$ when $y=3$?
I'm having trouble on this and I don't get why it's not $\frac{2\sqrt6\cdot15}{3}=10\sqrt6$?
Am I ...
0
votes
2answers
65 views
Prove the identity $\sqrt{17+2\sqrt{30}}-\sqrt{17-2\sqrt{30}}=2\sqrt{2}$
Prove the identity $$\sqrt{17+2\sqrt{30}}-\sqrt{17-2\sqrt{30}}=2\sqrt{2}.$$
We have $$\left(\sqrt{17+2\sqrt{30}}-\sqrt{17-2\sqrt{30}}\right)^2=17+2\sqrt{30}-2\sqrt{17+2\sqrt{30}}\cdot\sqrt{17-2\sqrt{...
1
vote
1answer
38 views
Find upper and lower bound
$\sqrt 3 = 1.73$ is correct to 3 significant figures. Find the upper and lower bounds of $$\frac{\sqrt 3-1}{2+\sqrt 3}$$ rounding your answers to 3 significant figures.
I'm confused here as:
UB of √...
1
vote
2answers
41 views
Show that $P=\sqrt{a^2-2ab+b^2}+\left(\frac{a}{\sqrt{a}-\sqrt{b}}-\sqrt{a}\right):\left(\frac{b\sqrt{a}}{a-\sqrt{ab}}+\sqrt{b}\right)$ is rational
Show that the number $$P=\sqrt{a^2-2ab+b^2}+\left(\dfrac{a}{\sqrt{a}-\sqrt{b}}-\sqrt{a}\right):\left(\dfrac{b\sqrt{a}}{a-\sqrt{ab}}+\sqrt{b}\right)$$ is a rational number ($P\in\mathbb{Q})$ if $a\in\...
3
votes
5answers
107 views
Show that $A=\sqrt{\left|40\sqrt2-57\right|}-\sqrt{\left|40\sqrt2+57\right|}$ is a whole number
Show that $A$ is a whole number: $$A=\sqrt{\left|40\sqrt2-57\right|}-\sqrt{\left|40\sqrt2+57\right|}.$$
I don't know if this is necessary, but we can compare $40\sqrt{2}$ and $57$: $$40\sqrt{2}\...
1
vote
1answer
46 views
Find the value of $A=\frac{3\sqrt{8+2\sqrt7}}{\sqrt{8-2\sqrt{7}}}-\frac{\sqrt{2\left(3+\sqrt7\right)}}{\sqrt{3-\sqrt7}}$
Find the value of $$A=\dfrac{3\sqrt{8+2\sqrt7}}{\sqrt{8-2\sqrt{7}}}-\dfrac{\sqrt{2\left(3+\sqrt7\right)}}{\sqrt{3-\sqrt7}}.$$
Since $$8+2\sqrt7=8+2\cdot1\cdot\sqrt{7}=1^2+2\cdot1\cdot\sqrt7+\left(\...
1
vote
1answer
85 views
Does $2x^5 - x^4 - 22x^3 - 23x^2 + 22x +24 = 0$ have exact solutions in radicals?
Does $2x^5 - x^4 - 22x^3 - 23x^2 + 22x +24 = 0$ have exact solutions in radicals?
A mysterious commenter said on Youtube this was the "easiest quintic equation of my life," and I'm ...
0
votes
3answers
50 views
Compare $\sqrt{6}-\sqrt{3}$ and $\sqrt{3}-\sqrt{2}$
Compare the numbers $a=\sqrt{5-2\sqrt{6}},b=\sqrt{6}-\sqrt{3}$ and $c=\sqrt{3}-\sqrt{2}$.
We have $$a=\sqrt{5-2\sqrt{2}\sqrt{3}}=\sqrt{\left(\sqrt{2}\right)^2-2\sqrt{2}\sqrt{3}+\left(\sqrt{3}\right)^2}...
0
votes
2answers
49 views
How do I find the minimum of a function? (Quadratic function with square root component)
I have a function that I am trying to find the minimum of. I know how to solve for the minimum of a quadratic function as well as a square root function. I don't know how to solve for the minimum of a ...
3
votes
6answers
64 views
In which of the intervals is $\sqrt{12}$
In which of the intervals is $\sqrt{12}:$
a) $(2.5;3);$
b) $(3;3.5);$
c) $(3.5;4);$
d)
$(4;4.5)$?
We can use a calculator and find that $\sqrt{12}\approx3.46$ so the correct answer is actually b. How ...
1
vote
2answers
42 views
Shifting decimals and square roots (eg, $\sqrt{64}$ vs $\sqrt{0.64}$ vs $\sqrt{6.4}$)
The square root of $64$ is $8$.
If the decimal of $64$ is shifted by two places to the left (i.e. $0.64$), then the decimal of the answer is shifted by one place to the left (i.e. $0.8$), so the ...
0
votes
0answers
18 views
Finding values of $x\in\mathbb{Z}$ that satisfy $f(x)\in\mathbb{Z}$ for a cubic
I am looking at octagonal numbers and wondering if there is any known trivial method to:
Given a function $f(x)$, find the integers $n$ such that $f(n)\in\mathbb{Z}$
I am sure simpler functions ...
3
votes
0answers
29 views
One or two roots of positive real numbers?
sorry for this basic question but I was just going through Rudin's Principles textbook, and it says in theorem 1.21 (p.10):
"$\textit{For every real $x>0$ and every integer $n>0$ there is ...
2
votes
2answers
111 views
Why is $\sqrt{6 + \sqrt{6 + \sqrt{6 + …}}} = 3$
This is a problem from SASMO Grade 8 (Secondary 2) Sample Questions.
Solve for $x$
$\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = 3$
Answer: $x=6$
I have tried this on a calculator: the more $x$ we add, the ...
1
vote
2answers
25 views
What am I doing wrong to calculate this standard deviation?
This post mentions in passing that the standard deviation of the data set (1, 50) is about 34.65. Wolfram Alpha confirms this, saying it is $\frac{49}{\sqrt{2}}\approx 34.648$. But when I try to ...
5
votes
1answer
78 views
Solving $(\sqrt{2})^x+(\sqrt{2})^{x-1}=2(2\sqrt{2}+1)$
I'm in stuck with this simple equation.
$$(\sqrt{2})^x+(\sqrt{2})^{x-1}=2(2\sqrt{2}+1)$$
This is my solution:
$$\begin{align}(\sqrt{2})^x+(\sqrt{2})^x(\sqrt{2})^{-1} &=4\sqrt{2}+2 \tag{1}\\[4pt]
2^...
1
vote
1answer
29 views
Please help me clear confusion over principal roots and identities for n-th radicals
From my old high school math textbook:
If ${a{\geq }0}$ and $n\in \mathbb{N} ^{\ast }$, then ${\sqrt[{n}] {a}}$ is the non-negative solution of ${{x}^{n}}=a$.
It then goes on to infer a number of ...
3
votes
0answers
127 views
Nonassociative algebra's closed under $\sqrt{}$?
Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients
$(a_0, \dots, ...
0
votes
1answer
74 views
Find the value of the expression: $\frac{1}{\sqrt{4}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{10}}+\cdots+\frac{1}{\sqrt{13}+\sqrt{16}}$
Find the value of the expression: $$\frac{1}{\sqrt{4}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{10}}+\cdots+\frac{1}{\sqrt{13}+\sqrt{16}}$$
After putting it into a calculator I worked out that it is equal to ...
1
vote
1answer
52 views
Are both values of cos15° that I obtained equal and if yes, why do I get such different-looking values?
I found the value of $\cos{15}^\circ$ using 2 methods.
Method 1: using $\cos{(a-b)} = \cos{a}\cos{b}+\sin{a}\sin{b}$
$\cos{(45-30)^\circ} = \cos{45^\circ}\cos{30^\circ}+\sin{45^\circ}\sin{30^\circ}$
$\...
1
vote
2answers
74 views
Sequence involving $\sqrt[n]{n}$ and $\text{log}$
I know that the sequence $\sqrt[n]{n}$ converges to 1 and that $\text{log}(\sqrt[n]{n})$ thus converges to 0 as $n\to\infty$ since the logarithmic function is continuous. But how can I calculate the ...
0
votes
1answer
46 views
How to calculate fractional roots on a computer? [closed]
I worked out a closed form formula for a function that is defined for positive integers $n$:
When $n$ is even: $x = 1 - (\alpha - 1)^n$
When $n$ is odd: $x = (\alpha - 1)^n + 1$
When this is ...
1
vote
1answer
40 views
Surd Manipulations.
If a=$-\sqrt {99}+\sqrt {999}+\sqrt {9999}$
b=$-\sqrt {99}-\sqrt {999}+\sqrt {9999}$
c=$-\sqrt {99}+\sqrt {999}-\sqrt {9999}$
Then $\displaystyle\sum_\limits{cyc} \frac{a^4}{(a-b)(a-c)}$ equals??
I ...
0
votes
1answer
38 views
radical and radical ideal
I am trying to get used to the terminology of commutative algebra. Let $R$ be a commutative ring with unity. Then I know radical $\sqrt{I} := \{ a\in R| a^n \in I, \textrm{for some $n \in \mathbb{Z}$} ...
1
vote
1answer
27 views
Meaning of the geometric mean in the power mean inequality
The power-mean inequality states that for positive real numbers $a_i$ and all real numbers $k_1,k_2$: $$(\frac{1}{n}\sum_{i=1}^{n}a_i^{k_1})^{\frac{1}{k_1}} \ge (\frac{1}{n}\sum_{i=1}^{n}a_i^{k_2})^{\...
0
votes
1answer
34 views
If $I[x]$ is a radical ideal of $R[x]$, then show that $I$ is a radical ideal of $R.$
Show that $I$ is a radical ideal of a commutative ring $R$ iff $I[x]$ is a radical ideal of $R[x].$
The problem is in the ''only if'' part.If $I[x]$ is a radical ideal of $R[x],$ then $I[x] = \sqrt J$ ...
1
vote
3answers
49 views
What is the expression $\sqrt{8\cdot32\cdot(-3)^2}$ equal to?
What is the expression $\sqrt{8\cdot32\cdot(-3)^2}$ equal to?
Sorry for the basic question. I am a little confused when solving such problems. They are very easy, I know, but still... Which is the ...
3
votes
1answer
98 views
Algorithm for dividing sums of square roots: $\tfrac{\sum_{i=1}^m \sqrt{x_i}}{\sum_{j=1}^n \sqrt{y_j}}$
As a part of a project for supporting symbolic computation of square roots (since using any sort of approximated calculation ends up accumulating errors after several iterations) I'm trying to ...
1
vote
1answer
53 views
In simplifying $\sqrt{\frac{(x^2 +x +3)^2}{(1-2q)^2}}$ to $\frac{x^2 +x +3}{|1-2q|}$, why use the absolute value?
I have seen in a question
$$\sqrt{\frac{(x^2 +x +3)^2}{(1-2q)^2}}$$ was given to be
$$\frac{x^2 +x +3}{|1-2q|}$$
Why was absolute value given to $1-2q$?
0
votes
0answers
40 views
Solve the equation in real numbers: $x-6+\frac{2}{\sqrt{x-2}}=\frac{1}{3}\log_3(\frac{x}{x^3+54})$
Solve the equation in real numbers: $x-6+\frac{2}{\sqrt{x-2}}=\frac{1}{3}\log_3(\frac{x}{x^3+54})$
My work: I have managed to find that $3$ is a solution to the problem.I tried to prove that this is ...
2
votes
3answers
83 views
Does the square root function behave the same when calculating complex roots?
This is incredibly basic, yet I cannot for the life of me confirm it. Let's say in the topic of complex numbers we are told to calculate the roots of:
$
z^4 = \sqrt 9
$
Are we to assume $ z^4 = 3 $ ...
1
vote
1answer
31 views
Is it possible to multiply out at least by $s^2$ in this root expression? $\sqrt{(a^2-4d)s^2+(4bd-4ac)s+4c^2}$
My goal is to pull out all the $s$ from under the square root.
$$\sqrt{(a^2-4d)s^2+(4bd-4ac)s+4c^2}$$
where $a,b,c,d$ are complex numbers and $s$ is a variable.
I tried solving for $s$, but that only ...
4
votes
2answers
105 views
Find all real numbers $x$ such that $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number
I want to find all values of $x\in \mathbb R$ such that the value of $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number.
I solved it as follows:
$x+2\sqrt{x}-1\ge 0$
$(\sqrt{x}+1)^2-2\ge 0$
$...
-1
votes
2answers
61 views
When I graph $y=x^{1/2}$ why does it only show the positive y values.
I understand the reason for $y=\sqrt{x}$. I've been told that the radical symbol gives out the positive answer. But $y=x^{1/2}$ doesn't use a radical symbol, and it still only shows the positive y ...
0
votes
1answer
37 views
If $f(x) = (x^2+2\alpha x + \alpha^2-1)^{\frac{1}{4}}$ has its domain and range so that union is $\mathbb{R}$, what does $\alpha$ satisfy?
My question asks:
If $f(x) = (x^2+2\alpha x + \alpha^2-1)^{\frac{1}{4}}$ has its domain and range such that their union is set of real numbers, what does $\alpha$ satisfy? (Answer is stated to be $\...
