Questions tagged [radicals]
For questions involving radical of numbers or radical of expressions (i.e. numbers/expressions raised to the power of a fraction).
3,121
questions
1
vote
2answers
62 views
Expressing $\sqrt{3 − \sqrt{5}}$ in the form $\frac{\sqrt{a}−\sqrt{b}}{c}$
Express $\sqrt{3 − \sqrt{5}}$ in the form $\frac{\sqrt{a}−\sqrt{b}}{c}$ for some integers $a,b,c$.
I'm not quite sure how to start this, can anyone give me a little hint or two
0
votes
1answer
42 views
Problems with the complex square root
I have a problem understanding the following procedure. (
It's from a script)
Consider the domain C[0,$\infty$) and the branch of the logarithm given by
$ log(z)=ln(|z|)+i \cdot arg(z)$ ,with $arg(z) \...
0
votes
1answer
26 views
Exponential Function: Q and n-th root? [duplicate]
I understand this function for $\mathbb{Z}$, however am left with questions on the Wiki definition with the $\mathbb{Q}$ domain:
$b^{p/q} = \sqrt[\leftroot{-2}\uproot{2}q]{b^p}$
How does the n-th root ...
-1
votes
2answers
41 views
How to simplify $\frac{\frac{\cos x}{2\sqrt{x}} + \sqrt{x}\sin x}{\cos^2x}$, step by step?
I want to simplify this expression
$$\frac{\frac{\cos(x)}{2\sqrt{x}} + \sqrt{x}\sin(x)}{\cos^2(x)}$$
and I know this is the answer
$$\frac{\cos(x)+2x \sin(x)}{2\sqrt{x}\cos^2(x)}$$
How can I get there ...
1
vote
1answer
29 views
what are the rules regarding $x,a,b$ for this expression to be true: $(x^a)^b = (x^b)^a$ (i am considering only for when $a,b$ are real)?
in particular i am asking for the case when one of the powers $a$ or $b$ is a fraction. in such a case, i believe the maths expression then may be ambiguous, as when you do to the power of a fraction,...
0
votes
3answers
51 views
how come square root of 2 times itself equals 2
since square root of $2$ is a irrational number, which we know or assume its decimal part is not a finite number, or doesn't terminate, how come we say that this infinite number (not in terms of being ...
0
votes
2answers
30 views
A simple question about nonnegativity of square roots [duplicate]
Say I have a sequence $(x_n)$ and also that $x_n \geq 0 $. Let $x = \lim_{n\to\infty}x_n$. I also know that $x \geq 0$. My question is,
Is $\sqrt{x_n} + \sqrt{x}$ a nonnegative quantity? I believe ...
17
votes
1answer
247 views
What algorithm is Newton using in the “De analysi” to extract the square root of a polynomial?
I'm reading a 1745 English translation of Newton's De analysi (apparently the most up-to-date there is, surprisingly). The Latin is here.
In this tract he shows how to use the integral power rule for ...
2
votes
3answers
68 views
How to simply this radical expression $\dfrac{\sqrt[3]{3}}{\sqrt[3]{1}+\sqrt[3]{2}}=\sqrt[3]{\sqrt[3]{2}-1}$
$$\dfrac{\sqrt[3]{3}}{\sqrt[3]{1}+\sqrt[3]{2}}=\sqrt[3]{\sqrt[3]{2}-1}$$
I could not multiply by the conjugate since it is a cube root. Can you show me a way to simplify it?
Thanks!
4
votes
2answers
105 views
Prove that $\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\leq 3+\sqrt{3}$
Problem. (Nguyễn Quốc Hưng) Let $0\le a,b,c\le 3;ab+bc+ca=3.$ Prove that $$\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\leq 3+\sqrt{3}$$
I have one solution but ugly, so I 'd like to find another. I will post my ...
0
votes
1answer
24 views
Solve for x Given Two Square Roots: Algebra Problem
I am trying to solve for all real numbers for $x$ given $5=\sqrt{9-x^2}+\sqrt{16-x^2}$.
The answer is that $x=\pm \frac{12}{5}$. I am looking for a clean way to do it. I am stumped, and I feel like I ...
1
vote
1answer
76 views
Finding the value of $\sqrt{z \sqrt{z \sqrt{z}}}…$
I was working on the following nested square root problem:
Let $a \in \mathbb R ^+$, what is the value of: $$\sqrt{a \sqrt{a \sqrt{a}}}...$$
I concluded that the answer is $a$ and then I thought ...
0
votes
0answers
41 views
How to precisely find the sum of a Beatty sequence
I am aware that, this has already been asked and answered on this question:
How to find $\sum_{i=1}^n\left\lfloor i\sqrt{2}\right\rfloor$ A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).
However, ...
1
vote
3answers
47 views
Why does $i^3$ equal $-i$ if you multiply the numbers inside a radical?
$i^3$ equals $-i$. Since $i$ is $\sqrt{-1}$, and you can multiply the number inside radicals that are being multiplied together, wouldn't $i^3$ equal $\sqrt{-1×-1×-1}$, which is $\sqrt{-1}$, which is $...
0
votes
1answer
50 views
Expressing $(5.8\sqrt{40} + 56.4) - (5.8\sqrt{10} + 56.4)$ in simplified radical form
I am trying to complete the question below, but I am not sure how to simplify the radical. What I have so far is
$$(5.8\sqrt{40} + 56.4) - (5.8\sqrt{10} + 56.4) \;=\; \text{the difference}$$
How does ...
2
votes
2answers
60 views
$n$-th Derivative $\frac{d^{n}}{d x^{n}} e^{-\sqrt{x} |\omega|}$ via Recursive Product Rule
Let $g(x) = x^{-\frac{1}{2}}$ and $f(x) = e^{-\sqrt{x} |\omega|}$. I am trying to find an expression for the $M$-th derivative of their product:
\begin{align}
\frac{d^M}{dx^M} \left[ f(x) g(x) \...
9
votes
1answer
290 views
A curious equality: Where do these numbers come from?
This identity, which was shared on math.stackexchange and seem curious at first sight, caught my attention. Here is the equality:
$$\color {red} {\dfrac{1646-736\sqrt{5}} {2641-1181\sqrt{5}} =\color{...
-4
votes
1answer
46 views
How does $\frac{3\sqrt{2x-1}-\frac{3x+4}{\sqrt{2x-1}}}{2x-1}$ simplify to $\frac{3x-7}{(2x-1)^{3/2}}$? [closed]
How does $$\frac{3\sqrt{2x-1}-\dfrac{3x+4}{\sqrt{2x-1}}}{2x-1}$$ simplify to $$\frac{3x-7}{(2x-1)^{3/2}}$$
-4
votes
3answers
51 views
why is $\sqrt{x^2} \ne x$? [closed]
If $\sqrt{3^2}= 3$, $\sqrt{2^2}\ne 2$.
why is $\sqrt{x^2}\ne x$?
2
votes
4answers
57 views
How to find the third root of a complex number by transforming the complex number into a root
I'd like to find all 3rd roots of this number z = i - 1. Now I've found formulas on how to do it; First we transform the complex number into this form
$$ \sqrt[n]{r} * e^{i\frac {\phi + 2k\pi}{n}} $$ ...
1
vote
0answers
34 views
Alternating Sum of Square roots of Binomial coefficients is always positive
Numerical experimentation seems to suggest that $$f_8(x) = \sqrt{1} - \sqrt{\binom{x}{1}} + \sqrt{\binom{x}{2}} - \sqrt{\binom{x}{3}} + \sqrt{\binom{x}{4}} - \sqrt{\binom{x}{5}} + \sqrt{\binom{x}{6}} -...
0
votes
0answers
57 views
I think I found a general formula for the square root of imaginary numbers. Is this correct?
I believe I found a formula for finding the square roots of any imaginary number. The formula is as follows:
$\pm \sqrt{ai} = \pm \sqrt{\frac{a}{2}}i \pm \sqrt{\frac{a}{2}}$
and here is my proof:
$\pm ...
0
votes
3answers
38 views
need help simplifying this radical $\sqrt[35]{128y^{42}}$
I am trying to figure out how to get to the solution below but have having difficulty. Can someone explain how to get to the solution.
$$\sqrt[35]{128y^{42}}$$
This is the answer but I can't figure ...
0
votes
4answers
63 views
Square root of 9 from a technical perspective.
When we say $$ \sqrt{9}= x $$
then $\;x = 3,\;$ right?
So why when we square both sides it becomes different:
$$ (\sqrt{9})^2 = x^2$$
$$9 = x^2$$
Here $\;x =\pm 3.$
So, does $\;x = \pm 3\;$ in $\sqrt{...
-1
votes
1answer
76 views
Simplifying $\sqrt{34+15\sqrt2}$ [closed]
$$\sqrt{34+15\sqrt2}$$
If we want $34+15\sqrt2$ to be a nice square $(a+b)^2=a^2+2ab+b^2$, most likely it is the case that $15\sqrt2$ corresponds to $2ab$. I don't know what to do from here. Is there ...
-2
votes
2answers
74 views
simplify $c=\sqrt{290-143\sqrt2}$ [closed]
I am trying to simplify $c=\sqrt{290-143\sqrt2}.$ I am solving a triangle and I got that $c^2=290-143\sqrt2.$ I have tried to use the formula for $\sqrt{a\pm\sqrt{b}}$ but it seemed useless at the end....
4
votes
6answers
216 views
$\sqrt{x^2+12y}+\sqrt{y^2+12x}=33$ subject to $x+y=23$
Solve the system of equations:
$\sqrt{x^2+12y}+\sqrt{y^2+12x}=33$, $x+y=23$
The obvious way to solve it is by substituting for one variable. However I was looking for a more clever solution and went ...
1
vote
1answer
30 views
In relation to the Babylonian method for computing a square root, if N/b = c, will c always be less than b?
I'm trying to wrap my head around the Babylonian method/algorithm for computing the square root of a number N. I can't seem to explain in words why, when a is too large i.e a^2 > N, then why you ...
-1
votes
1answer
59 views
What are the values of $a$ and $b$?
I got this question from our quiz wrong I wonder how my teacher got the correct answers shown. I tried solving the first question but didn't get the right answers. I know the formula for the surface ...
3
votes
1answer
70 views
Extremely accurate fractions for square roots
Approximation
The following is a simple and amazingly accurate way to get a rational approximation to square roots:
To find $\sqrt n$, guess a fraction $p/q$ near $\sqrt n$. (So $nq^2 \approx p^2$).
...
-2
votes
1answer
23 views
Give an example of a proper ideal I of a ring R such that the radical of I is equal to R i.e. √I=R.
I am having trouble with finding such example. In commutative case with identity such example does not exist as 1∈R=√I implies 1∈I and so I=R. Is there any in noncommutative case or a ring without ...
0
votes
1answer
51 views
How to read $\sqrt[\frac{1}{x}]{n}$
How do I read $$\sqrt[\frac{1}{x}]{n}$$
I was curious if this was equal to $n^x$. This may sound a weirdly obvious question and instinctively I thought the answer was yes (since a root is a fractional ...
0
votes
1answer
61 views
why is $\sqrt{16}$ not equal to $-4$? [closed]
$(-4)^2=4^2=16$
I wonder why $\sqrt{16}$ is not equal to $-4$.
2
votes
0answers
68 views
Simplifying a radical-trigonometric expression for the hendecagon angle
This question is related to my very first question on this site, on constructing the hendecagon. The Gleason paper I referred to states the following identities, which lead to constructions of a ...
0
votes
0answers
36 views
Squaring the equation of $4$ variables
I have an equation of the following form
\begin{align}
A + B x + C y + D z = (E x + F y + G z) t
\end{align}
where $x, y, z, t$ are square roots of some other term and rest coefficients are constant. ...
4
votes
0answers
122 views
Infinite product representation of $\sqrt{1-z}$
Does there exist an infinite sequence of complex numbers $\{ z_i \}_{i=1,2,\cdots}$ such that
$$ \boxed{ \sqrt{1-z} = \prod_{i=1}^\infty \left( 1 - \frac{z}{z_i} \right) \qquad \textrm{(for } |z|<1)...
-1
votes
1answer
54 views
Under what condition, it is allowed to use this equality $f(n)=\sqrt{f(n)}\sqrt{f(n)}$? [closed]
I have a function $f(n)$ which is positive for odd $n$ and negative for even $n$. My question is, am I allowed to write $f(n)$ as $\sqrt{f(n)}\sqrt{f(n)}\quad$ for all $n\quad$? Or, this is allowed ...
1
vote
1answer
44 views
Find natural numbers $u, v$ for which $\displaystyle 2\sqrt{u} + 4\sqrt{v - u} - 7 \geq v$.
As I said in the title, the problem states:
Solve the following inequation in $\mathbb{N}^2$:
$$ 2\sqrt{u} + 4\sqrt{v - u} - 7 \geq v $$
Source: Mathematical challenges for $8^{th}$ grade.
Approach: ...
0
votes
1answer
16 views
Comparing pixels for color difference - taking Alpha into account
Say I want to compare pixels to see how similar in color they are. This wikipedia article describes how:
https://en.wikipedia.org/wiki/Color_difference
it says:
My question is, there not just a R,G,B ...
3
votes
2answers
119 views
Finding a closed form for the following integral: $\int_0^1\sqrt{1+x^k}dx$
Is there a nice closed form, for the following integral:
$$\int_0^1\sqrt{1+x^k}dx$$
And how can I derive it?
I have no idea how to get started, thanks for any help. This problem came up when I was ...
0
votes
1answer
46 views
How to simplify $x^{x^{-5}(5)}=5^{-\frac{25}{\sqrt[5]{5^{16}}}}$ and find a value from it?
The problem is as follows:
First simplify:
$$x^{x^{-5}(5)}=5^{-\frac{25}{\sqrt[5]{5^{16}}}}$$
Then using $x$ find: $x^{-50}$
The alternatives in my book are as follows:
$\begin{array}{ll}
1.&5\\
2....
1
vote
1answer
27 views
How to estimate the temperature of Keelung when it is modeled by a radical?
The problem is as follows:
The expression from below represents a model made by satellite based on the weather patters in Keelung
$$T=\sqrt[81^{3^{n}}]{\left[\sqrt[3]{8^{3^{3^{n+1}}}}\right]^{3^{3^{n}}...
0
votes
2answers
105 views
Limit of square root [closed]
Can you help me to calculate this limit as '$a$' varies in $\mathbb{R}$:
$$\lim _{x\rightarrow +\infty} \sqrt{2x^2 + x + 1} - ax$$
-1
votes
1answer
123 views
Finding The Value Of Infinite Nested Radicals
The problem is to find the value of,
$\sqrt{-1+1\sqrt{-2+2\sqrt{-3+3\sqrt{-4+4\sqrt{...}}}}}$
Even though I have solved the problems which have a definite pattern which repeats itself and we make some ...
0
votes
3answers
84 views
Solve $\frac{7}{x+\sqrt{x+5}}+\frac{7}{x-\sqrt{x+5}}=8$
Solve the equation: $$\dfrac{7}{x+\sqrt{x+5}}+\dfrac{7}{x-\sqrt{x+5}}=8.$$ I am not sure how to approach the problem. Should we first determine the domain? I think we can also check for every value we ...
0
votes
2answers
33 views
1
vote
3answers
139 views
Integer values of $\sqrt{n}+\sqrt{n+2005}$ where $n \in \mathbb{Z}$
Find integer values of $\sqrt{n}+\sqrt{n+2005}$ where $n$ is an integer.
So far, I have just listed some squares which are larger than $2005.$ The first few are $2025,2116, 2209,2304,$ etc. I can just ...
6
votes
1answer
82 views
General term of a sequence and its limit.
Having a sequence $\;\sqrt[3]5\;,\;\sqrt[3]{5\sqrt[3]5}\;,\;\sqrt[3]{5\sqrt[3]{5\sqrt[3]5}}\;,\;\ldots\;,\;$ what could be the general term and its limit ?
Note that the second term is cube root of $5$...
1
vote
1answer
105 views
Analytic continuation of square root with branch cut along the negative imaginary axis by using a suitable logarithmic branch.
We are working with real integrals and the complex residue theorem. In order to solve the following integral:
$$ \int_0 ^\infty \frac{\sqrt{x}}{1+x^2},$$
I will have to think about how a square root ...
5
votes
1answer
50 views
How to simplify the fraction?
How to simplify the fraction $ \displaystyle \frac{\sqrt{3}+1-\sqrt{6}}{2\sqrt{2}-\sqrt{6}+\sqrt{3}+1} $ to $ (\sqrt{2}-1)(2-\sqrt{3}) $?
I've checked it in the calculator and both give the same ...
