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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2
votes
1answer
40 views

For $x\geq 0$, is $\sqrt{x}$ the magnitude of $x^{1/2}$?

Motivation \begin{align} 4^{1/2} &= \begin{cases} \left(2^2\right)^{1/2}\\ \left(\left(-2\right)^2\right)^{1/2} \end{cases} \\ &= \begin{cases} 2\\ ...
2
votes
2answers
40 views

In simplifying the formula that I've derived for finding the square root of a complex number to the standard formula.

So by easy means, I derived $\sqrt{a+ib} = \sqrt{\frac{a(a+1)+b^2}{2}}+i(\frac{b}{\sqrt{2}\sqrt{a(a+1)+b^2}})$ But then I checked for the actual formula it is this; $\sqrt{a+ib} = \sqrt{\frac{\...
3
votes
1answer
58 views

$n$-th derivative of $x^\alpha$ where $\alpha = m + 1/2$

It is well-known that, for any real $\alpha$ and nonnegative integer $n$ $$ \frac{d^n x^\alpha}{dx^n} = \alpha(\alpha-1)\cdots(\alpha - n + 1) x^{\alpha - n} $$ I just found out that the coefficient ...
0
votes
0answers
11 views

How to construct a function whose inverse branches are related by a radical?

Consider a function $f(x)$ that has a minimum $c$ and whose inverse around that minimum has two branches, $x_1(f)$ and $x_ {2}(f)$. I look for functions $f$ such that $\frac{1}{x_1(f)}-\frac{1}{x_2(f)...
1
vote
3answers
67 views

Simplify $\sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}} $

Simplify $$ \sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}} $$ Attempt: $$ \sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}} = \sqrt{10 + 2\sqrt{6} + 2\sqrt{10} + 2\sqrt{15}} = \sqrt{10 + 2(\sqrt{6} + \...
3
votes
4answers
118 views

Simplify $ \frac{ \sqrt[3]{16} - 1}{ \sqrt[3]{27} + \sqrt[3]{4} + \sqrt[3]{2}} $

Simplify $$ \frac{ \sqrt[3]{16} - 1}{ \sqrt[3]{27} + \sqrt[3]{4} + \sqrt[3]{2}} $$ Attempt: $$ \frac{ \sqrt[3]{16} - 1}{3 + \sqrt[3]{4} + \sqrt[3]{2}} = \frac{ \sqrt[3]{16} - 1}{ (3 + \sqrt[3]{4}) +...
2
votes
0answers
59 views

Denesting infinite nested radicals: $\sqrt{1+ \sqrt{2+ \sqrt{4 + \sqrt{8 + \sqrt{\dots}}}}}$ [duplicate]

I tried to denest the infinite nested radical $$\sqrt{1+ \sqrt{2+ \sqrt{4 + \sqrt{8 + \sqrt{\dots}}}}} \; .$$ My first try was to find a functional equation for $$\alpha(x) = \sqrt{1+ \sqrt{x+ \...
-2
votes
0answers
20 views

Where can I end up with this formula [closed]

$z = \frac{M_0^{3/4}}{F_0^{1/2}} \sqrt{1-\sqrt{(1-\frac{F_0t}{M_0})^3}}$ What would be the value of $z$ if $M_0$ is ignored(insignificant). I know it might be silly but the result I got is very far ...
1
vote
4answers
141 views

Find $y$ in $\sqrt{4+(y-6)^2}+\sqrt{16+(y-3)^2}=3\cdot\sqrt{5}$ [closed]

I would like to solve this equation for $y$. Any tips? It seems like you cant really do it analytically? $$\sqrt{4+(y-6)^2}+\sqrt{16+(y-3)^2}=3\cdot\sqrt{5}$$
2
votes
3answers
66 views

Find “A” in this equation.

$$ \sqrt[3] {A-15√3} + \sqrt[3] {A+15√3} = 4 $$ Find "A" ? The way of exponentiation took too much time, is there any easier method?
2
votes
3answers
80 views

Doubt with $\sqrt[3]{x} \ne x^{\frac{1}{3}}$

In a test, there was the following question: What is the value of $(-0.125)^{\frac{1}{3}}$? One of the possible answers was "$-0.5$" and another answer was "None of the above". It is important to ...
1
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0answers
35 views

When is the Galois group of a quintic not $S_5$ if we use this particular method?

I think that I am misunderstanding something fundamental about the technique used to decide if higher order polynomials are solvable by radicals using Galois theory. If we have a cubic it's not to bad ...
0
votes
2answers
91 views

Evaluating $\lim_{x \to 2}\frac{\sqrt{3x-5}-\sqrt[3]{x-1}}{x-2}$ [closed]

Find the limit: $$\lim_{x \to 2}\frac{\sqrt{3x-5}-\sqrt[3]{x-1}}{x-2}$$ I tried multiplying by the conjugate both numerator and denominator, but it wasn't helpful. Any ideas?
-2
votes
4answers
86 views

Why is $1=\sqrt{(-1)(-1)}=\sqrt{i^2\cdot i^2}=\sqrt{i^4}=\sqrt{\left(e^{i\frac{\pi}{2}}\right)^4}=\sqrt{e^{i2\pi}}=e^{i\pi}=-1$ wrong?

Why is $1=\sqrt{(-1)(-1)}=\sqrt{i^2\cdot i^2}=\sqrt{i^4}=\sqrt{\left(e^{i\frac{\pi}{2}}\right)^4}=\sqrt{e^{i2\pi}}=e^{i\pi}=-1$ wrong? my friend showed me this "proof" - what's the first step that ...
2
votes
2answers
80 views

Find all natural numbers $n$ such that $\sqrt{n-1}+\sqrt{n+1}$ is rational… [duplicate]

There aren't any natural number $n$ for which the given condition is satisfied. Here is how I proved it: For $\sqrt{n-1}+\sqrt{n+1}$ to be rational, $\sqrt{n-1}$ and $\sqrt{n+1}$ must individually be ...
4
votes
3answers
148 views

Calculating the nth super-root when n is greater than 2?

Tetration (literally "4th operator iteration") is iterated exponents, much like how exponents are iterated multiplying. For example, $2$^^$3$ is the same as $2^{2^2}$, which is $4^2$ or 16. This ...
0
votes
4answers
72 views

Prove that $\lim_{n \to \infty}\sqrt{n} \cdot \left(\sqrt{n+1} - \sqrt{n}\right)=\frac12$. [duplicate]

This is a very unspecific and maybe stupid question, so I apologize for that. We recently had an exam that I failed, because I had pretty much no time to practice before that. Now I got to learn all ...
1
vote
3answers
115 views

Can i factor this expression: $x^3+y^3+z^3$

I have the following numerical expression, which is exactly equal to $1$ Text version: ...
0
votes
0answers
33 views

Equivalence between square roots and vector

Can a square root be equated to a vector? My idea is that the square root of -1 is equal to i. Since i is connected to the (complex) plane, a geometrical structure, then a more 'profound' equality ...
2
votes
1answer
42 views

Integral of signum and a root

I need to evaluate (or at least find an upper bound) of this type of integrals: $$ \int_0 ^{\frac{1}{2}} \text{sign}\left(\frac{1}{2} - x\right) \frac{1}{\left(\frac{1}{2}-x\right)^s} \text{d}x \,,$$ ...
1
vote
0answers
45 views

Calculating the values of $S(e^{-\frac{\pi}{25}})$ and $S(e^{-5\pi})$.

Are the values I calculated correct? $S(q)=-R(-q)$ where R(q) is the Rogers-Ramanujan continued fraction. $R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$ ...
5
votes
3answers
106 views

When going from $(x+2)^2=5$ to $x+2=\pm \sqrt{5}$, why isn't there also a $\pm(x+2)$?

Say I am solving the following equation: $$(x+2)^2 = 5$$ $$x + 2 = \pm \sqrt{5}$$ $$x = -2 \pm \sqrt{5}$$ However, when I took the positive and negative square root of $5$ in the second line, I ...
0
votes
1answer
14 views

Total Cost According to Miles Driven [closed]

A computer repair person charges 50.00 dollars per hour, plus an additional mileage fee. The charge for mileage varies directly with the square root of the number of miles traveled. If one hour plus ...
3
votes
4answers
76 views

Evaluating $\sqrt{a\pm bi\sqrt c}$

I recently encountered this problem $$\sqrt{10-4i\sqrt{6}}$$ To witch I set the solution equal to $a+bi$ squaring both sides leaves $${10-4i\sqrt{6}}=a^2-b^2+2abi$$ Obviously $a^2-b^2=10$ and $2abi=-...
2
votes
3answers
108 views

Given $x-\sqrt {\dfrac {8}{x}}=9$, what is $x-\sqrt{8x}$ equal to?

Given $x-\sqrt {\dfrac {8}{x}}=9$, what is $x-\sqrt{8x}$ equal to? My attempt: We have \begin{align} x-\sqrt {\dfrac {8}{x}}=9 \implies -\sqrt {\dfrac {8}{x}}=9-x \implies \dfrac {8}{x}=(9-x)^2 \end{...
0
votes
1answer
22 views

Questioning the rules of radicals

Does an absolute value symbol need to be included every time a radical with a variable expression has an even index? I understand that in some cases there needs to be a absolute value symbol just in ...
1
vote
3answers
63 views

Question on Radicals [closed]

Please how do I simplify this: Rationalisation seems daunting because of the cube root. Any suggestion will be appreciated
0
votes
1answer
48 views

roots: square roots, cube roots, x^1/4, x^1/5 [duplicate]

I was having confusion over the concepts of roots. My confusion begins as: I know that if $x^{1/n}=y$ then $y^n=x$ must be true. Case 1: With square roots, let's take $y=4^{1/2}$ We know $y=2$ is a ...
3
votes
3answers
105 views

Find the minimum of the function $y=\sqrt{-x^2+4x+21}+\sqrt{-x^2+3x+10}.$

Find the minimum of the function $$y=\sqrt{-x^2+4x+21}+\sqrt{-x^2+3x+10}.$$ By computer I found $\min_y=3;$ then I will prove $y\ge 3$. After squaring we got $$(x+2)(178-37x)\ge 0\quad \forall -2\le ...
0
votes
1answer
70 views

Dungeons & Dragons 3.5 Experience

I built a spreadsheet that accurately determines a character's level by looking at the experience. You can find a chart of this here: https://www.ign.com/wikis/dungeons-and-dragons/...
3
votes
6answers
107 views

Is $x=-2$ a solution of the equation $\sqrt{2-x}=x$?

Solve the equation: $$\sqrt{2-x}=x$$ Squaring we get $$x^2+x-2=0$$ So $x=1$ and $x=-2$ But when $x=-2$ we get $$\sqrt{4}=-2$$ But according to algebra $$\sqrt{x^2}=|x|$$ So is $x=-2$ invalid?...
2
votes
4answers
69 views

What is the square root of $(-5)\cdot(-5)$ and how is it different from $\sqrt{(-5)^2}$?

What is the square root of $(-5)\cdot(-5)$ and how is it different from $\sqrt{(-5)^2}$? Can anybody explain?
2
votes
2answers
113 views

Compute the integral $\int\limits_0^1 \frac{3x}{\sqrt{4-3x^2}} dx $?

I am struggling to compute the following equation. \begin{equation} \displaystyle\int_0^1 \dfrac{3x}{\sqrt{4-3x^2}} dx \end{equation} We are expected to use u-substitution, but I'm stuck and ...
1
vote
2answers
101 views

Square root of matrix product

My linear algebra abilities are somewhat limited, so this may be a very basic question. Suppose we have two positive definite matrices A and B. Is $(AB)^{1/2}=A^{1/2}B^{1/2}$? Furthermore, is $(AB^{-1}...
1
vote
2answers
46 views

Simple question about pointing out my mistake in denesting this radical

Could someone please help me spot my mistake here : I want to denest $\sqrt{1+\sqrt{2}}$, I did the following $\sqrt{1+\sqrt{2}}=\sqrt{d}+\sqrt{e}$ Squaring both sides gives $1+\sqrt{2}=d+e+2\sqrt{...
0
votes
2answers
25 views

Equivalency of two radical expressions proof

I know that $$\sqrt{2+2\sqrt{2}}-\sqrt{1+\sqrt{2}}$$ is equivalent to $$\sqrt{\sqrt{2}-1}.$$ However, I do not know how to prove that one is equal to the other and vice versa. The cause that I want to ...
8
votes
1answer
262 views

$S=\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+\dots+\sqrt{m}$ is almost an integer. Find $m$

For an integers $m$ and $n$, $1<m\le n$ , we need to find the best $m$ so that $S=\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+\dots+\sqrt{m}$ is almost an integer. Example: when $n=40$, then the best ...
5
votes
3answers
83 views

Rationalize the denominator of $\frac{\sqrt{7\sqrt{3}+4\sqrt{5}}}{\sqrt{7\sqrt{3}-4\sqrt{5}}}$

I have to rationalize the denominator of $A = \frac{\sqrt{7\sqrt{3}+4\sqrt{5}}}{\sqrt{7\sqrt{3}-4\sqrt{5}}}$. I multiplied the fraction by $\frac{\sqrt{7\sqrt{3}-4\sqrt{5}}}{\sqrt{7\sqrt{3}-4\sqrt{5}}...
4
votes
2answers
132 views

Is taking square root of both parts of equation in this way is an equivalent transformation of the equation?

Solve equation $$(2x+7)^2=(2x-1)^2$$ $t=2x-1 $, so equation becomes $$ (t+8)^2 = t^2 $$ Now let's make a "prohibited" - take a square root from both parts (minding that $\sqrt{x^2} = \lvert x\...
0
votes
3answers
50 views

Constraints of $\frac{\sqrt{x^3}-\sqrt{y^3}+\sqrt{x^2y}-\sqrt{xy^2}}{\sqrt{x}+\sqrt{y}}$

We have the expression A= $\frac{\sqrt{x^3}-\sqrt{y^3}+\sqrt{x^2y}-\sqrt{xy^2}}{\sqrt{x}+\sqrt{y}}$. I have to simplify it. First I want to define the constraints of $x$ and $y$ but I have some ...
1
vote
4answers
46 views

Simplify $\frac{a+b-\sqrt{a^2-b^2}}{\sqrt{a+b}}$

I have to simplify the following expression: $A =\frac{a+b-\sqrt{a^2-b^2}}{\sqrt{a+b}}$ Answer: $\sqrt{a+b}-\sqrt{a-b}$ I am trying to find the constraints of $a$ and $b$. I think that $a^2-b^2 \...
0
votes
1answer
24 views

Simplify $D=\sqrt{x^2}+\sqrt{x^2-2x+1}$

I have this expression, and I have to simplify it: $D=\sqrt{x^2}+\sqrt{x^2-2x+1}$ Answer: $1)$ If $x<0, D=1-2x$, $2)$ if $0\le x<1, D=1;$ $3)$ if $x>1, D=2x-1$. I received $|x-1| + |x|$...
2
votes
1answer
42 views

Simplify $B=\sqrt{x^2} - x$

I have to simplify the following expression: $B=\sqrt{x^2} - x$ The only thing that I can do is: $\sqrt{a^2}=|a|$, thus $B=|x|-x$. Is that enough?
3
votes
7answers
87 views

Why can I simplify radicals? (eg, $\sqrt{153} = \sqrt{3}\cdot \sqrt{3}\cdot \sqrt{17}$)

I know it might sound like a ridiculously easy question to answer, but I just can't put two and two together for some reason. Say for example you have: $$\sqrt{153}$$ You can break it down to $$\...
1
vote
2answers
49 views

Solve$(10+6\sqrt3)^{\frac{1}{3}}-(-10+6\sqrt3)^{\frac{1}{3}}$

We need to solve the following equation $y=(10+6\sqrt3)^{\frac{1}{3}}-(-10+6\sqrt3)^{\frac{1}{3}}$ and it is equal to 2 while I am getting the value in excel I am not able to solve it manually ...
4
votes
5answers
123 views

Derivative when $(\sqrt{x})^2$ is involved?

Problem: If $f(x)=\frac{1}{x^2+1}$ and $g(x)=\sqrt{x}$, then what is the derivative of $f(g(x))$? My book says the answer is $-(x+1)^{-2}$. This answer seems flawed because $(\sqrt{x})^2$ is being ...
3
votes
8answers
75 views

Radical equation solve $\sqrt{3x+7}-\sqrt{x+2}=1$. Cannot arrive at solution $x=-2$

I am to solve $\sqrt{3x+7}-\sqrt{x+2}=1$ and the solution is provided as -2. Since this is a radical equation with 2 radicals, I followed suggested textbook steps of isolating each radical and ...
1
vote
4answers
97 views

Find $ \lim_{x\to-\infty}\ln\left(\sqrt{x^2+4}+x\right)$.

Find $$ \lim_{x\to-\infty}\ln\left(\sqrt{x^2+4}+x\right). $$ I got this limit which gives me $\ln(0^+)=-\infty$. Is this ok? I ended up with my answer in this way: my limit is equal to $$\ln\left(\...
-3
votes
2answers
48 views

How do I solve this question? I need help. [closed]

I can't understand how to solve this question. Please help me. $\left[3({x^{1/3} - x^{-1/3}})\right]^{1/3} = 2,\; \text{then}\; x^{1/3} + x^{-1/3} = ?$
0
votes
0answers
29 views

Summation of Radical n

How can I calculate this Summation based on $n$ ? $$S = \sum_{k=1}^\infty\sqrt[2^k]{n}$$