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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0
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1answer
44 views

The fundamental aspects of the square root [closed]

When I was in High School learning algebra we came upon solving for roots. When doing this for a quadratic you sometimes end up having square roots in your answer. Due to uncertainty we cannot ...
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2answers
53 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 ...
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1answer
35 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 $\...
7
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0answers
73 views

Sufficient & necessary conditions for comparing sums of square roots: $\sum_{i=0}^m \sqrt{x_i} \ge \sum_{j=0}^n \sqrt{y_j}$

I'm trying to implement an algorithm for comparing sums of square roots, i.e. to find out whether one sum is greater than/equal to the other $$ \sum\limits_{i=1}^{m} \sqrt{x_i} \ge \sum\limits_{j=1}^{...
-1
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3answers
106 views

Transforming a function by square root [closed]

When $f(x) = x/2 - 8$ is transformed into $y = \sqrt{f(x)}$, there are 2 invariant points, $(a, b)$ and $(c, d)$. If $d>b$, what is $d$? I have no idea how to tackle this question.
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2answers
40 views

If $\sqrt{9^x - 6^x} = \sqrt{6^x - 4^x}$, then the number of values of $x$ are:

If $\sqrt{9^x - 6^x} = \sqrt{6^x - 4^x}$, then the number of values of $x$ are: I have solved the above like this: $\sqrt{9^x - 6^x} = \sqrt{6^x - 4^x}$ Squaring: $9^x - 6^x = 6^x - 4^x$ $9^x + 4^x = ...
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0answers
81 views

To find when $\sqrt{\frac{7^{n}+1}{2}}$ is prime

Find all $n$ such that $\sqrt{\frac{7^{n}+1}{2}}$ is a prime. I have tried elementary methods like factorising and using $\pmod{7}$ and so on, but with no luck. I think this question could be ...
5
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2answers
156 views

How to remove roots from an equation?

The question here is how (if it is even possible) to remove the square root terms and transform the following equation to a polynomial with one unknown $x$. The coefficients $a$, $b$, $c$, and $d$ are ...
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1answer
40 views

Square Root of $9a + 36\sqrt{3ax} + 108x$ [closed]

An old mathematics book found on Google asserts that: Square root of $9a + 36\sqrt{3ax} + 108x$ Equals to $3\sqrt{a} + 6\sqrt{3x}$ I find this incorrect, but am not 100% sure. My derivation is: Let $m ...
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2answers
77 views

Expressing $\sqrt{3 − \sqrt{5}}$ in the form $\frac{\sqrt{a}−\sqrt{b}}{c}$ [closed]

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
43 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
54 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
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2answers
31 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
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1answer
263 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
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3answers
73 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
113 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
25 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 ...
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2answers
62 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
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1answer
293 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
48 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
62 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
35 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
58 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
64 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
77 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
217 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
60 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 ...
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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 ...
3
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4answers
106 views

How to solve $\lim_{n \to \infty}\frac{1}{\sqrt[3]{n^3+n+1}-\sqrt{n^2-n+2}}$ without L'Hopital?

$\lim_{n \to \infty}\frac{1}{\sqrt[3]{n^3+n+1}-\sqrt{n^2-n+2}}$ $\lim_{n \to \infty}\frac{1}{\sqrt[6]{(n^3+n+1)^2}-\sqrt[6]{(n^2-n+2)^3}}$ but because this limit is still the type of $\frac{1}{\infty-\...
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. ...
4
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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
45 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
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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
votes
2answers
120 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 ...

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