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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Radicals in a Cyclotomic Field Extension

I was looking for the theorems which describe about all the kinds of radicals contained in a cyclotomic extension. With radical I mean the number, say $x$, is not in $\mathbb{Q}$ but some power, say $...
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1 answer
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Some property of ideals

We have that $I$ is an ideal in ring $R$. How prove: if $x^a \in I$ and $y^b \in I$ show that $(x+y)^{(a+b)} \in I$? I don't have any idea. I know what is it ideal
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1 answer
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Is there a way to ensure that a simplified square root is right?

I was reading this article about square root and they simplify $\sqrt{75}$ to $5\sqrt{3}$. Is there a way to ensure that the answer is correct, going from $5\sqrt{3}$ to $\sqrt{75}$? For example, I ...
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0 votes
2 answers
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The modulus of a complex number being negative?

Why can't the modulus of a complex number considered negative? I mean it is obvious but has it ever been considered to invent any new maths in order to analyze such cases similarly to what created ...
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-2 votes
1 answer
87 views

How does $\dfrac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}}$ become $1$? [closed]

$$\dfrac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}}$$ The answer to this question is "1" but I have no idea how !! Please show the steps to solve the problem.
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0 votes
0 answers
31 views

What is the proof for the $(\sqrt[n]{a})^n$ equaling $a$ or $|a|$ if $n$ is odd or even?

If I use $64$ and $-64$ as the radicands, and have $2$ or $3$ as the indices, I know that it all works out arithmetically (except for $\sqrt[2]{-64}$ which has no real solutions) but I'm not sure how ...
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2 votes
3 answers
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Is $a+5^{1/3}b+5^{2/3}c$ a root of any cubic polynomial in $\mathbb{Q}$?

For arbitrary $a, b, c \in \mathbb{Q}$, let $w := a + 5^{1/3} b + 5^{2/3} c$, is $w$ a root of any cubic polynomial in $\mathbb{Q}$? I guess the cubic polynomial always exists. But I am confused about ...
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1 vote
1 answer
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Fractional Exponent Identity: Is the inside of a root necessarily calculated first?

I know $a^{b/c}=\sqrt[c]{a^b}$, but does $a^b$ have to be calculated first, or does $a^{b/c}=(\sqrt[c]a\,)^b$ also work? Thanks.
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0 votes
0 answers
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Every nonnegative real analytic function is the square of a real analytic function

Let $f:\mathbb R\to \mathbb R$ an analytic nonnegative real function that may have zeros. There always exists an analytical real function $g:\mathbb R\to \mathbb R$ such that $f =g^2$? If $f>0$ ...
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6 votes
3 answers
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Solving $\sqrt{x+3}+\sqrt{5-x}-2\sqrt{15+2x-x^2}=-4$

Solve the following equation: $\sqrt{x+3}+\sqrt{5-x}-2\sqrt{15+2x-x^2}=-4$ Since real solutions are to be found, the domain of $x$ is $[-3; 5]$. I immediately found that $15+2x-x^2$ can be factored ...
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  • 445
7 votes
4 answers
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Value of $x^6 - 2\sqrt{3}x^5 - x^4 + x^3 - 4x^2 + 2x - \sqrt{3}$ given $x = \frac{1}{2-\sqrt{3}}$?

It is given that $x = \frac{1}{2-\sqrt{3}}$. Find the value of $x^6 - 2\sqrt{3}x^5 - x^4 + x^3 - 4x^2 + 2x - \sqrt{3}$. Well I tried rationalising and I came to know that $x = 2 + \sqrt{3}$. And I ...
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5 votes
0 answers
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Is there an element-free proof that preimages of radical ideals are radical?

Suppose we have a ring homomorphism $\phi:A\rightarrow B$ and an ideal $J\subseteq B$. I just spent way too much time on an exercise in commutative algebra, because the element-free definition of the ...
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2 votes
3 answers
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Denesting radicals $\sqrt[3]{-22+15\sqrt[3]{3}+9\sqrt[3]{9}}$ and $\sqrt[3]{8-9\sqrt[3]{3}+3\sqrt[3]{9}}$

I am trying to do denesting radicals:$$\sqrt[3]{-22+15\sqrt[3]{3}+9\sqrt[3]{9}}$$ and $$\sqrt[3]{8-9\sqrt[3]{3}+3\sqrt[3]{9}}$$ I tried to find Ramanujan polynomial like this link denesting radicals ...
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1 vote
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Ambiguous solution involving radicals

Evaluating the expression: $$\sqrt{5+2\sqrt{6}}+ \sqrt{5-2\sqrt{6}} $$ So my solution involves rewriting the expression inside the bigger radical signs as perfect squares: $$\sqrt{\left(\sqrt{3}+\sqrt{...
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  • 131
0 votes
2 answers
70 views

How to take the square root of $BC$ where $BC^2$ equals $56-32\sqrt3$ [duplicate]

$BC^2$ equals $56-32\sqrt3$ what is the square root of BC? The dimensions I used to get this far (and I confirmed that they are correct) are A,B,C, in a right triangle, where B is $\sqrt{(42-24\sqrt3)}...
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3 votes
1 answer
56 views

Why is $\pm\sqrt{(11y-8)^2} = \pm(11y-8)$? - Analytic Geometry

$3x^2-7xy-6y^2-2x+17y-5=0$ My original goal here was to know whether or not this was a degenerate conic, so I isolated $x$ by applying the Quadratic Formula. $3x^2+(-7y-2)x+(-6y^2+17y-5)=0$ $x = \frac{...
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1 vote
2 answers
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What is $\sqrt{-1}$ (or $\sqrt{-j}$) in the Hyperbolic (Split-complex) Numbers?

Given a number system such that $j^2 = 1, j \ne \pm 1$, what would be the solution to $z^2 + 1 = 0$? Are the hyperbolic numbers not closed under taking roots unlike the complex numbers? My assumption ...
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1 vote
1 answer
38 views

The degree of simple radical extensions

Let $m \le n$ be positive integers. Does there necessarily exist a field extension $K/F$ such that $[K:F] = m$ and $K = F(u)$ for some $u \in K$ satisfying $u^{n} \in F$? In other words, given $m \le ...
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14 votes
5 answers
322 views

Prove that $\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{b^2+c^2}}+\frac{c}{\sqrt{c^2+d^2}}+\frac{d}{\sqrt{d^2+a^2}}\leq3$

Prove that if $a,b,c,d$ are positive reals we have: $$\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{b^2+c^2}}+\frac{c}{\sqrt{c^2+d^2}}+\frac{d}{\sqrt{d^2+a^2}}\leq3.$$ I think that I have found a equality ...
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  • 944
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1 answer
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How do I simplify the absolute value of $\sqrt{1-it}$ for any arbitrary $t\in\mathbb R$?

So the question is in the title itself: How do I simplify the absolute value of $\sqrt{1-\iota t}$ for any arbitrary $t\in\mathbb R$? I would have replaced $\iota$ by $-\iota$, multiplied by it, and ...
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2 votes
2 answers
198 views

Understanding a 'geometrical proof' of irrationality of √2

I had been having trouble understanding a proof of the irrational nature of √2. I found this proof in the first page of the foreward to 17 theorem provers of the world where a 'geometrical proof' (is ...
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  • 121
1 vote
1 answer
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Harmonic addition theorem [duplicate]

In order to satisfy the following equality: $$a\cos{\theta}+b\sin{\theta}=r\cos{\left({x-x_0}\right)}$$ $x_0 = \tan^{-1}{\frac{b}{a}},$ and $a^2 + b^2 = r^2$. The latter statement implies $r = \pm\...
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  • 367
4 votes
3 answers
129 views

Simplify $\sqrt[3]{9\sqrt3-11\sqrt2}$

Simplify $$\sqrt[3]{9\sqrt3-11\sqrt2}$$ How can we actually simplify this radical?
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-2 votes
1 answer
50 views

Why is an irrational denominator used for the volume of a tetrahedron? [closed]

Volume of a tetrahedron: $$V = \frac{a^3}{6\sqrt 2}$$ Where $a$ is the edge length. Why is the radical $\sqrt 2$ left in the denominator? Reworking $$V = \frac {\sqrt 2a^3}{6×2} $$to get $$V = \frac {...
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0 answers
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Confusion between exponential functions e.g. $10^x$ and $e^x$

Assume we we have the following exponential function: $10^x$ If we think about it across the x-axis in $0$ the height of the $y$ axis is $1$ and where $x = 1$ it is $10$. Now if we split the $[0,1]$ ...
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  • 1,551
0 votes
1 answer
81 views

Why is $\lim\limits_{x \to \infty} \frac{(\sqrt{x})!}{(\sqrt{x+1})!}$ = 1 and not 0?

Wouldn't we expect it to be equal to 0? I suspect that $\lim\limits_{x \to \infty} \frac{1}{\sqrt{x+1}}$ should be equal to $\lim\limits_{x \to \infty} \frac{(\sqrt{x})!}{(\sqrt{x+1})!}$. And the ...
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6 votes
1 answer
157 views

How is this cube root continued fraction generated?

I'm looking at the examples on the Wiki Page Generalized Continued Fractions . The introduction to general root-finding states a formula: But then, looking at the second example for cube root of 2, ...
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3 votes
2 answers
67 views

Range of $\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - a)^2}$ on $0 \le a,$ $b,$ $c,$ $d \le 1.$

Let $0 \le a,$ $b,$ $c,$ $d \le 1.$ Find the possible values of the expression $$\sqrt{a^2 + (1 - b)^2} + \sqrt{b^2 + (1 - c)^2} + \sqrt{c^2 + (1 - d)^2} + \sqrt{d^2 + (1 - a)^2}.$$ I tried to use ...
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4 votes
1 answer
32 views

Functions $f(x)=\log_x(a)$ and $f(x)=\sqrt[x]{a}$

I was refreshing my memory on the power, exponential, nth-root, and logarithmic functions when I realised that there is a whole pair of functions missing. If I know the exponent, I can make $y=x^a$ or ...
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2 votes
0 answers
76 views

Simplified the cube root of the complex number

I have simplified this cube root: $$ \sqrt[3]{1+i}=\sqrt[6]{2}\bigl(\frac{\sqrt{6} + \sqrt{2}}{4} +{\frac{\sqrt{6} - \sqrt{2}}{4} }i\bigr)$$ which is simplified algebraic expression form. Now, I was ...
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1 vote
1 answer
105 views

Simplifying $\sqrt{1+\sqrt{5}(6-2\sqrt{5})^{1/4}}$

I have the radical $$\sqrt{1+\sqrt{5}(6-2\sqrt{5})^{1/4}}$$ for exam preparation (middle school): I need to simplify it in natural numbers. My attempt is: We know the rule: $(a-b)^2=a^2-2ab+b^2$ Let'...
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0 votes
0 answers
42 views

Trigonometric integral with square roots

When studying a fixed-point problem for a certain integral equation which arises in my research, I am led to evaluating an integral of the form $\int_0^{\pi} e^{-\sqrt{a^2-\cos t}} \cos t\,dt$ where $...
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0 votes
0 answers
23 views

Should there be an absolute value inside a radical after simplifying the radicals power?

I am trying to help my sibling with math but their teacher seems to want an absolute value inside the radical when simplifying the radical's power, not just outside, which some teachers care and some ...
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0 votes
0 answers
41 views

Notation for nth root... can n be non-integer? [duplicate]

We write the nth root of $x$ as $\sqrt[n]{x}$. Can $n$ be non-positive integer? Does the notation $\sqrt[\sqrt{2}]{2}$ or $\sqrt[-3]{8}$ make sense?
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6 votes
1 answer
551 views

Correlation between $\sqrt{1/10}$ and length of powers of integers.

I'm searching information about this simple problem involving square roots and length of powers. It's very simple but it seems interesting, at least for me. I'm not a mathematician. Description By ...
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  • 171
1 vote
4 answers
106 views

Proving that $\sqrt{x^2+1}+x>0$ for all $x$

Recently while dealing with inverse hyperbolic functions, I came across the expression $$\sinh^{-1}x=\ln(x+\sqrt{x^2+1})$$ We know that $f(x)=\sinh^{-1}x$ is defined for all real values of $x$ since ...
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  • 636
0 votes
1 answer
57 views

Rationalizing higher index roots than cubic, such as $\frac{1}{\sqrt[10]{2}-1}$

so..., someone in here asked about how to rationalize higher index roots, they sad something about the telescoping identity, but I don't get it. I know there is another way to do it and has something ...
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0 votes
3 answers
56 views

How to find derivative in radical function?

I need to find derivatives of following functions: $$\frac32x^\frac32-\frac{2x^2}{3}$$ $$-2\sqrt{x}-\frac{-2}{\sqrt{x}}$$ So starting from first one, I have tried to first simplify the fractions to ...
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3 votes
1 answer
80 views

$ \sqrt a+\sqrt b +\sqrt c = 1$ Prove: $\frac{a^2+bc}{\sqrt{2a^2(b+c)}}+\frac{b^2 + ac}{\sqrt{2b^2(a + c)}}+\frac{c^2 +ab}{\sqrt{2c^2(a+b)}}\geq1$ [duplicate]

I'm having some trouble with this problem: a, b, c are positive real numbers where $ \sqrt a + \sqrt b + \sqrt c = 1 $ . Prove the following inequality: $$\frac{a^2 + bc}{\sqrt{2a^2(b+c)}} + \frac{b^2 ...
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  • 31
5 votes
2 answers
146 views

Why isn't $\sqrt{64x^4y^8z^6}$ equal to $8x^2y^4z^3$?

I simplified $$\sqrt{64x^4y^8z^6}$$ by taking the square root of $64$ (getting $8$), $x^4$ (getting $x^2$), $y^8$ (getting $y^4$), and $z^6$ (getting $z^3$). My answer, $8x^2y^4z^3$, not quite right ...
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2 votes
2 answers
66 views

Determine if a number n is a power

What would be an efficient algorithm to determine if $n \in \mathbb{N}$ can be written as $n = a^b$ for some $a,b \in \mathbb{N}, b>1$? So far, I've tried: ...
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  • 29
0 votes
0 answers
42 views

When does squaring the square root of a function give me the absolute value of the function

Why does $[\sqrt{f(x)}]^2=f(x) $ , but $\sqrt{g^2(x)}=|g(x)|$ ?
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-1 votes
1 answer
64 views

Find closest integer to $(3+\sqrt7)^4$ by hand [duplicate]

Find the closest integer to $$(3+\sqrt7)^4$$ by hand, without knowing the correct value of $\sqrt7$ (Maybe just knowing that $2<\sqrt7<3$). My work:$$(3+\sqrt7)^4 = (16+6\sqrt7)^2 = 508+192\...
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  • 773
0 votes
1 answer
72 views

When converting to parametric equation, should plus-minus sign be used?

I'm following a tutorial which says this: Convert the following equation to a pair of parametric equations for $x$ and $y$ in terms of $t$: $$y=x^2+3$$ Step 1 - Set $t$ equal to $x^2$: $$t=x^2$$ Step ...
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0 votes
2 answers
67 views

Does the nth root of $n \cdot 2^{3n}+3^{2n}$ converge as $n \to \infty$?

Consider the sequence $a_n = \sqrt[n]{n \cdot 2^{3n}+3^{2n}}$, $n\in\mathbb{N}$. With a string of inequalities, one can show that $a_n$ is bounded and the graph of the function $ f(x) = \sqrt[x]{x \...
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  • 465
-1 votes
2 answers
131 views

Please help me solve this radical.

Please show me how does $\sqrt{12\sqrt[3]{2} - 15}$ + $\sqrt{12\sqrt[3]{4} - 12}$ = 3. I can't find the answer anywhere else. The answer in the book also says $$\left(\sqrt{12\sqrt[3]{2} - 15} + \sqrt{...
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1 vote
3 answers
110 views

Evaluation of $\sqrt[3]{40+11\sqrt{13}}+\sqrt[3]{40-11\sqrt{13}}$

Evaluate $$\sqrt[3]{40+11\sqrt{13}}+\sqrt[3]{40-11\sqrt{13}}$$ The solution is $5$. Suppose $\sqrt[3]{40+11\sqrt{13}}=A, \sqrt[3]{40-11\sqrt{13}}=B$ We have $$A^3+B^3=80, A^3-B^3=22\sqrt{13}$$ Two ...
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  • 773
0 votes
1 answer
81 views

$\sqrt{\sin^{2}(\frac{\alpha}{2})}=\sin(\frac{\alpha}{2})$?

Question: Two forces $\vec{P}$ and $\vec{Q}$ acting at a point have a resultant $\vec{R}$ and the resolved part of $\vec{R}$ along $\vec{P}$ is equal to the magnitude of $\vec{Q}$. Prove that the ...
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2 votes
2 answers
72 views

Does closure under Euclidean distance imply closure under square roots (in this set)?

Suppose you start with the number 1 (and 0 because why not) and the set is closed so that you may add, subtract (smaller from bigger), multiply, and divide (except 0) any elements of the set. It is ...
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  • 73
2 votes
2 answers
73 views

find the $x$ of quadratic equation, such that the squre root of quadratic equation result is integer

given this equation $$S=\sqrt{x^2+1500x-1472}$$ find $x$, such that $x$ is positive integer, and $S$ is positive integer. I have tried to solve this, and I get that $x = 36$, but I get that $x$ by ...
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