For questions involving radical numbers or expressions (i.e. expressions which involve $\sqrt[n]{\text{something}}$).

learn more… | top users | synonyms

0
votes
1answer
63 views

Is there any simple analytic method for solving $\sqrt{x}+y=7$ and $x+\sqrt{y}=11$ simultaneously. [duplicate]

I am thinking of a nice and simple analytic method to solve the following equations simultaneously: $$\sqrt x+y=7;\\x+\sqrt y=11.$$ To my suprise I can't. But, I solve the system numerically using ...
2
votes
2answers
28 views

Taking root from absolute expression

Why is the following true? (Where all terms are positive) $$|x-y| < \epsilon^2 \implies |\sqrt x - \sqrt y| < \epsilon$$
5
votes
5answers
100 views

How $\sqrt{2}=1+\frac{1}{\sqrt{2}+1}$?

I have found it in the chapter about chain fractionals. I am unable to transform it to such state. $$\sqrt{2}=1+\sqrt{2}-1=?=1+\frac{1}{\sqrt{2}+1}$$
7
votes
6answers
531 views

How to solve the inequality $x^2>10$ using square roots?

Solve the inequality: $$x^2>10$$ How am I supposed to do this? It doesn't make sense when I take into account that if $x^2=10$ then $x=+\sqrt{10}$ and $x=-\sqrt{10}$ But how am I supposed to ...
3
votes
0answers
81 views

Simplifying $\sqrt[3]{a\pm\sqrt{b}}$

Let $$x=\sqrt{a\pm\sqrt{b}}$$ We know that $$x=\sqrt{\frac{a+\sqrt{a^2-b}}{2}}\pm\sqrt{\frac{a-\sqrt{a^2-b}}{2}}$$ But, what about cubic root? Let $$y=\sqrt[3]{a\pm\sqrt{b}}$$ Is there any formula to ...
1
vote
2answers
37 views

Radical Inequality

$\sqrt{2x-1}$ + $\sqrt{3x-2}$ > $\sqrt{4x-3}$ + $\sqrt{5x-4}$ I have attempted to solve this by squaring each side, resulting in $5x + 2\sqrt{2x-1}\sqrt{3x-2} - 3 > 9x + 2\sqrt{(4x-3)(5x-4)} - 7 ...
2
votes
2answers
62 views

Are numbers like $\left ( -2 \right )^{\sqrt{2}}$ real or complex?

I know that numbers with rational power can be converted to radicals and based on the degree of the radical we can say that whether they are real or complex. But what about numbers like $\left ( -2 ...
3
votes
3answers
68 views

Explaining why $\sqrt {x^2+a} = x\sqrt{1+ \frac{a}{x^2}}$ For $x>0$.

I understand the technical operation of extracting $x^2$ out of the root, but is there a way proving it? $$\sqrt {x^2+a} = x\sqrt{1+ \frac{a}{x^2}}$$
13
votes
3answers
226 views

How to prove: $\left(\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt[4]{25}-\sqrt[4]{125}}}-1\right)^{4}=5$?

Question: show that: the beautiful ${\tt sqrt}$-identity: $$ \left({2 \over \sqrt{\vphantom{\Large A}\, 4\ -\ 3\,\sqrt[4]{\,5\,}\ +\ 2\,\sqrt[4]{\,25\,}\ - \,\sqrt[4]{\,125\,}\,}\,}\ -\ ...
0
votes
2answers
62 views

Simplify and Combine radicals/exponents [closed]

How do I simplify the following: $$x^{-\frac12}+x^{-\frac32}\;?$$ Can it be factored and simplified? Or combined into one fraction? I'm confused because of the powers of xbeing different negative ...
0
votes
2answers
54 views

For what integer values of $m$ and $n$ is $\frac{4m-n}{n}$ a rational square?

Question for what integer values of $m$ and $n$ with $(m,n)=1$ is $\frac{4m-n}{n}$ a rational square? Note the motivation for this question is a curiosity i noticed, that the smallest angle of the ...
2
votes
2answers
37 views

Solving $\frac{\sqrt{108x^{10}}}{\sqrt{2x}}$

Simplify $$\frac{\sqrt{108x^{10}}}{\sqrt{2x}}$$ $\dfrac{\sqrt{108x^{10}}}{\sqrt{2x}}= \dfrac{(108x^{10})^{1/2}}{(2x)^{1/2}}$ The $1/2$ exponent cancels $\implies \dfrac{108x^{10}}{2x}$ ...
0
votes
9answers
219 views

Difference between $\sqrt{x^2}$ and $(\sqrt{x})^2$

According to my logic, $$\large\sqrt{x^2} = x^{2\times \frac{1}{2}} = x = x^{\frac{1}{2}\times 2}={(\sqrt{x})}^2$$ But when I look at the graphs of these guys, they're totally different. Edit: ...
2
votes
4answers
50 views

Convert $8 {\sqrt 2}$ to $ \frac{16} {\sqrt 2}$

Can anyone please explain to me how this happens!? My brain cannot think how to get from one to the other. Thanks! $8 {\sqrt 2}$ to $ \frac{16} {\sqrt 2}$
3
votes
4answers
110 views

Deeply confused about $\sqrt[5]{a^5}=(a^5)^{1/5}$

So is this correct? $\sqrt[5]{a^5} = \left(a^5\right)^{\frac{1}{5}}$ I need proof why $\left(a^5\right)^\frac{1}{5}$ can or cannot just be $a^\frac{5}{5}$ or just $a$? I think of that rule of ...
1
vote
4answers
89 views

How many solutions $k>1$ does the equation $\exp ((k-1)/( k+1))=\sqrt{k}$ have?

I have the following equation: $e^{\frac{k-1}{k+1}}=\sqrt{k}$. The question is: how many solutions does it have? ($e$ is Euler's constant and k is a positive real number >1).
1
vote
1answer
42 views

Simplification of expressions with radicals in Maple

Having for example the expression $$\frac{abc\sqrt2}{d\sqrt{ab}}$$ (which results from a sequence of manipulations), can I force Maple to write it in the form $$\frac{c\sqrt{2ab}}{d}.$$ Many might ...
3
votes
4answers
97 views

Why does $\dfrac{8}{\frac{8\sqrt{145}}{145}} = \sqrt{145}$?

I can't seem to work out why this is true: $$\frac{8}{\dfrac{8\sqrt{145}}{145}} = \sqrt{145}$$ Could someone break it down for me?
0
votes
3answers
61 views

Can anyone help me understand the simplification of $\frac{\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2}\;$?

Can anyone help me understand the following simplification of the fraction? $$\dfrac{\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2} = 5 + 2\sqrt 6$$ I cant understand how to simplify the left-hand side to get ...
5
votes
3answers
128 views

Find the value of this infinitely nested radical (it appears to obtain multiple values)

Find the value of $$\sqrt{1-\sqrt{\frac{17}{16}-\sqrt{1-\sqrt{\frac{17}{16}-\cdots}}}}$$ This is not as simple as it looks for one reason - there are $2$ real solutions to the equation ...
3
votes
1answer
53 views

Simplify $\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} + \frac {\sqrt {45}}{2}$

I am trying to find the value of: $$\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} + \frac {\sqrt {45}}{2}$$ I have the key with the answer $\sqrt 5$ but am wondering how I can easily get to that ...
4
votes
2answers
77 views

How prove that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $

How check that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $?
0
votes
2answers
50 views

Simplifying Surd Fractions

can someone show me how to simple surd fractions such as: $$\frac{{8\sqrt 3 }}{2}$$ Can someone please help me here?
2
votes
3answers
92 views

$\ \sqrt{x+39}-\sqrt{x+7}=4 $

So I tried to solve this problem for x $\ \sqrt{x+39}-\sqrt{x+7}=4 $ I multiplied both sides ($\ \sqrt{m}\cdot\sqrt{n}=\sqrt{mn} $) $\ (\sqrt{x+39}-\sqrt{x+7})^2=16 $ $\ ...
3
votes
1answer
67 views

A problem from Komal

For every integer $n\ge 2$ let $$P(n)=\prod (\pm \sqrt{1} \pm \sqrt{2} \cdots \pm \sqrt{n})$$ where the product is over all possible permutations of the signs. Prove $P(n)\in \mathbb{Z}\;\forall ...
1
vote
5answers
40 views

$12\frac{\sin 45^\circ}{\sin 60^\circ}$ Need help breaking this down.

Otherwise known as $12\dfrac{\left(\frac{1}{\sqrt2}\right)}{\left(\frac{\sqrt3}{2}\right)}$ How do you simplify this multi level fractional radical expression into $4\sqrt{6}$.
1
vote
1answer
25 views

Find the speed S by using radical equations

There are two word problems that I cannot write as radical equations. 1.A formula that is used for finding the speed s, in mph, that a car was going from the length L, in feet, of its skid marks can ...
3
votes
1answer
113 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
1
vote
1answer
54 views

How do you find the following limit as x approaches infinity?

$\lim_{x\to \infty} \sqrt{x^2+9} - \sqrt{x^2-2}$ I have tried multiplying by the conjugate but the square roots are throwing me off and I'm not sure what to do next. How do you solve this?
7
votes
1answer
83 views

Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $

If $x\in\mathbb{R}$ find the maximum value of $$ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $$ I tried this: Let $$y= \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$ For maxima ...
0
votes
1answer
21 views

Help inverting a non-linear system of equations

I have a set of two equations like this $$ \gamma_3=\left(\frac{1}{\sqrt{1+2c_3^2+6c_4^2}}\right)^3 \left( \alpha_1\,c_3^3 + \alpha_2\,c_3c_4^2 + \alpha_3\,c_3c_4 + \alpha_4\,c_4\right)\\ ...
1
vote
2answers
108 views

Proof in Rudin, real numbers

There is one thing in this proof I do not get. Why can he say that $k < y$?
7
votes
1answer
67 views

Nested radicals and n-th roots

There are many beautiful infinite radical equations, some relatively straightforward, some much more subtle: $$ x = \sqrt{ x \sqrt{ x \sqrt{ x \sqrt{ \cdots } } } } $$ $$ \sqrt{2} = \sqrt{ 2/2 + ...
2
votes
2answers
97 views

How to find a cube root of numbers?

While, I was solving a problem of Chemistry [Solid State] when I encountered an equation like : $$a^3 = 3.612 \times 10^{-23} $$ Where, a is just a quantity [Actually, it is the length of a cubic ...
2
votes
1answer
26 views

Why is the formula for generating Van der Corput sequences called an Inverse Radical Function?

The Van de Corput sequence can be generated using the following formula: $\phi_b(n) = \sum_{i=1}^N { a_j \over b^{i-1}}.$ where this can be defined as the "one-dimensional sequence defined by the ...
2
votes
0answers
64 views

Find the value of $\sqrt{1+2\sqrt{1+3\sqrt{1+…}}}$ [duplicate]

Find the value of $$\sqrt{1+2\sqrt{1+3\sqrt{1+....}}}$$ I have done a similar question before in which only a single number is involved, for example $\sqrt{2+2\sqrt{2+2\sqrt{2+....}}}$ which ...
0
votes
1answer
132 views

Deradicalization of denominators

Task: Develop a fraction equivalent to $$ 1\over{\sum\limits_{i=0}^{n-1}c_in^{i/n}} $$ in which the denominator is rational.
6
votes
5answers
202 views

Simple solving Skanavi book exercise: $\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}$

Simple way to solve this exercise $$ x = \sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}} $$
0
votes
3answers
27 views

Semicircle - Sketching Points

I am having problems understanding how to sketch/solve $x = \sqrt{1 - (y-1)^2}$. Please, any help is much appreciated.
5
votes
3answers
117 views

Proving an expression is $4$

Is $\displaystyle\sqrt[\huge3]{\frac{1}{2} \left(56-\sqrt{\frac{84640}{27}}\right)}+\sqrt[\huge 3]{\frac{1}{2} \left(\sqrt{\frac{84640}{27}}+56\right)}=4$ true ? This was asked during an oral ...
0
votes
0answers
13 views

How to find the value or transform it? [duplicate]

For $x>1$, $\underset{n\rightarrow\infty}{lim}(2x)^{\frac{n}{2}}\underset{\text{n square roots} }{\underbrace{\sqrt{x-\sqrt{x(x-1)+\sqrt{x(x-1)+\sqrt{x(x-1)+...}}}}}}=?$
1
vote
2answers
26 views

Direct proportion - sum of numbers and square root of sum of the squared numbers

Is it true for all the cases that if $x + y > a+b$, then $\sqrt{x^2 + y^2} > \sqrt{a^2 + b^2}$? In other words - is there a direct proportion between sum of numbers and square root of the sum of ...
1
vote
0answers
95 views

What does “radical cube zero” mean?

My hobby is taking comics way too seriously. And I just came across a math topic. In a certain comic (Fantastic Four 51, according to some polls the greatest comic issue ever) there's a machine for ...
6
votes
6answers
425 views

Why does $\sqrt{n\sqrt{n\sqrt{n \ldots}}} = n$?

Ok, so I've been playing around with radical graphs and such lately, and I discovered that if the nth x = √(1st x √ 2nd x ... √nth x); Then $$\text{the ...
3
votes
0answers
78 views

patterns in the decimal expansions of adjacent square and cube roots

For fun I made a table in Excel which evaluated the square and cube roots of whole numbers in ascending order. Then of the result, I extracted the first, second and third decimal place digits, then ...
1
vote
1answer
58 views

Solving equations with multiple roots

So I was bored and tackled a bonus exercise during math class, and I managed to derive that the solution could be found as the solution for $x$ in the following equation: ...
10
votes
6answers
2k views

Can the square root of a real number be negative?

Can the square root of a real number be negative? Dealing with the questions of functions in eleventh class my maths teacher says that square root of a real number is always positive. How is it ...
0
votes
3answers
32 views

simultaneous equations with irrational variables

Solve the simultaneous equations $a\sqrt a+b\sqrt b=183$ and $a\sqrt b+b\sqrt a=182$ I made an attempt in vain to equate the coefficients and eliminate
3
votes
2answers
72 views

Cubic polynomial - radical expression of roots

Let $f=X^3+X^2-2X-1$ be a polynomial with the three roots $x_1,x_2,x_3$ with $x_1=2\text{cos}(\frac{2 \pi}{7})$. We define $z:=(x_1-x_2)(x_1-x_3)(x_2-x_3)$. I want to find a radical expression for ...
0
votes
1answer
33 views

Char(F)=0, K/F is abelian and F contains a primitive root of unity

$\mathrm{Char}(F)=0, K/F$ is abelian . Let $n$ be a positive integer such that $f^n=1$ for every $f$ in $G(K/F)$ and $F$ contains a primitive $n^{th}$ root of unity. Prove there exist $x_1, x_2, …, ...