Questions tagged [radicals]

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

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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
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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) \...
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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 ...
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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
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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,...
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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 ...
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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 ...
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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 ...
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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!
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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 ...
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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 ...
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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 ...
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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, ...
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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 $...
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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
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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) \...
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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{...
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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 $$\frac{3x-7}{(2x-1)^{3/2}}$$
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3answers
51 views

why is $\sqrt{x^2} \ne x$? [closed]

If $\sqrt{3^2}= 3$, $\sqrt{2^2}\ne 2$. why is $\sqrt{x^2}\ne x$?
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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 $$ \sqrt[n]{r} * e^{i\frac {\phi + 2k\pi}{n}} $$ ...
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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}} -...
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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 ...
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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 ...
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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{...
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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 ...
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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....
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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
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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 ...
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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
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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 $nq^2 \approx p^2$). ...
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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
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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
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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
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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
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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. ...
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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)...
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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
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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: ...
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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
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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
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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
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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}}...
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2answers
105 views

Limit of square root [closed]

Can you help me to calculate this limit as '$a$' varies in $\mathbb{R}$: $$\lim _{x\rightarrow +\infty} \sqrt{2x^2 + x + 1} - ax$$
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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
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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
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2answers
33 views

Reduce the radical:

From: Lumbreras Editors So I proceeded: $ \left(\sqrt{\sqrt{2+...
1
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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
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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
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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
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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 ...

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