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Questions tagged [radicals]

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

8
votes
2answers
300 views

Comparing the magnitudes of expressions of surds

I recently tackled some questions on maths-challenge / maths-aptitude papers where the task was to order various expressions made up of surds (without a calculator, obviously). I found myself ...
0
votes
2answers
40 views

Prove the following inequalities for any positive real numbers x,y

Given the following inequality, I'm finding it difficult to demonstrate that it holds for all real numbers: $xy^3$ $\leq \frac{1}{4}x^4 + \frac{3}{4}y^4$ I have normalized the equation to avoid ...
0
votes
1answer
71 views

Prove the inequality $\sqrt{\frac{a^2+1}{b+c}}+\sqrt{\frac{b^2+1}{a+c}}+\sqrt{\frac{c^2+1}{a+b}}\ge 3$

Let $a;b;c\in R^+$ such that $ab+bc+ca>0$. Prove that $$\sqrt{\frac{a^2+1}{b+c}}+\sqrt{\frac{b^2+1}{a+c}}+\sqrt{\frac{c^2+1}{a+b}}\ge 3$$ I have seen the similar question is $$\frac{a^2+1}{b+c}+\...
0
votes
1answer
17 views

If I assume that the variables are positive number, should I put absolute value when taking them of out of a square root?

If I assume that the variables are positive number, should I put absolute value when taking them of out of a square root? like if I assumed that u is a positive number and I have square root u to the ...
-1
votes
4answers
57 views
-3
votes
1answer
39 views

How to add square roots with different radicals? [on hold]

I am stuck on a math problem and I don't even know how to approach it. $$ (-2\sqrt{2}+4\sqrt{3})(\sqrt{5}-3\sqrt{3})$$ Please help.
3
votes
3answers
93 views

Graphing Without using Calculus $f(x) = \sqrt{x + 2} - \sqrt{x - 2}$

I am trying to solve the following problem: I can visualize how it looks like "approximately", it's essentially something like $\sqrt{x - 2} - 2$, with the difference that it increases faster. But ...
17
votes
13answers
3k views

Calculating the square root of 2

Since $\sqrt{2}$ is irrational, is there a way to compute the first 20 digits of it? What I have done so far I started the first digit decimal of the $\sqrt{2}$ by calculating iteratively so that ...
0
votes
1answer
27 views

Calculating complex numbers with my TI-84+

I got a problem wrong and I don't understand how. The question was to compute: $\sqrt{-1875.5+9.68i}$ I said this would be equal to .112+43.31i because that's the number my calculator gives me. ...
1
vote
1answer
39 views

If $(-24, 12)$ is a point on the graph of the function $y=f(x)$, identify one point on the graph of function $y = -2 \sqrt{f( -(x - 2)) - 4} + 6$.

I am fairly confident I know the process to solving this equation. But the answer I am given in my textbook for the new $y$ value confuses me. Here is a brief overview my steps: $(-24, 12)$ is the ...
0
votes
1answer
106 views

What is the maximal value of $\sqrt[3]{1+(x-y)}+\sqrt[3]{1+4(y-z)}+\sqrt[3]{1+9(z-x)}$?

what is the maximal value of $\sqrt[3]{1+(x-y)}+\sqrt[3]{1+4(y-z)}+\sqrt[3]{1+9(z-x)}$? I think we will use Holder inequality , try to transform the expression before using the inequality
2
votes
3answers
82 views

Inequality$\frac{(x+y)^{1/2} + (y+z)^{1/2} + (x+z)^{1/2}}{{(x+y+z)}^{1/2}} \leq 6^{1/2}$

I am not sure what approach to take. Any ideas or hints, pls? $$\frac{(x+y)^{1/2} + (y+z)^{1/2} + (x+z)^{1/2}}{{(x+y+z)}^{1/2}} \leq 6^{1/2}$$
1
vote
2answers
42 views

How to find integers $p$ and $q$ such that $(p\sqrt{2}+q)^2=34-24\sqrt{2}$

Find integers $p$ and $q$ such that $(p\sqrt{2}+q)^2=34-24\sqrt{2}$. I approached this question first by expanding the the left-hand side to get: $$2p^2 +2\sqrt{2}pq+q^2 = 34-24\sqrt{2}$$ The ...
0
votes
0answers
37 views

Why is $\sqrt3$ not an element of $\mathbb Q$? [duplicate]

I'm bad at maths. Why is $\sqrt3$ not an element of $\mathbb Q$? (A book states this)
2
votes
2answers
53 views

Domain for index of Radical Sign

What is the domain of $\sqrt[x]{a}$, and is $\sqrt[x]{a}=a^{1/x}$ always true?? I was told that the domain of $\sqrt[x]{a}$ is natural numbers and the domain of $a^{1/x}$ is real numbers, so they are ...
7
votes
2answers
206 views

inquality $\sum_{i=1}^{n}\frac{\sqrt{x_{i}-x_{i-1}}}{x_{i}}\le\sum_{i=1}^{n^2}\frac{1}{i}-\frac{1}{2}$

let $x_{i}\in N^{+}$,and such $1=x_{0}\le x_{1}\le x_{2}\le\cdots\le x_{n}$ show that $$\sum_{i=1}^{n}\dfrac{\sqrt{x_{i}-x_{i-1}}}{x_{i}}\le\sum_{i=1}^{n^2}\dfrac{1}{i}-\dfrac{1}{2}$$ maybe can use C-...
0
votes
3answers
33 views

Finding the area of an isosceles triangle inscribed in a square

I've now been going around and around trying to solve this problem, and I just haven't managed. It's supposed to have a quick and smart way of solving, and the solution must be 6. Below, the problem ...
1
vote
0answers
38 views

Why is $\operatorname{rect}(\frac{1}{2}-\sqrt{w}) = \operatorname{rect}(w-\frac{1}{2})$

I have to determine the first order probability density function for $w(t, \epsilon)$, with the hypothesis that I have two statistically independent functions $x(t, \epsilon)$ and $y(t, \epsilon)$ ...
0
votes
1answer
35 views

Calculus limit with sum of radicals [duplicate]

I am trying to calculate the following limit, but to be honest I've been failing miserably. $\lim\limits_{n\rightarrow\infty}\frac{\sqrt{1+\frac{1}{n}}+\sqrt{1+\frac{2}{n}}+...+\sqrt{1+\frac{n}{n}}}{...
0
votes
3answers
37 views

Solving complex algebraic equation with radicals

How do I solve this math question? If $x$ and $y$ are rational numbers and $(3 + 4\sqrt{3})(x + y\sqrt{3}) = 26$, find the sum of $x$ and $y$. I tried solving for $x$ and $y$ individually to add them ...
2
votes
7answers
3k views

Algebra mistake related to the equation $2\sqrt x=x-2$ [closed]

The original equation is $2\sqrt x=x-2$ and I replaced $x$ with $4-2\sqrt3$. I am not sure what I did wrong with the algebra. Could someone please help me. My work is posted below.
1
vote
3answers
64 views

Prove that $| xy-\sqrt{(1-x^2)(1-y^2)}|\leq1$ where $|x|\leq1$ and $|y|\leq1$ [duplicate]

Prove that $| xy-\sqrt{(1-x^2)(1-y^2)}|\leq1$ where $|x|\leq1$ and $|y|\leq1$ I tried: $x=\sin\alpha$ and $y=\cos\beta$ $\sqrt{(1-x^2)(1-y^2)}=\sqrt{\cos^2\alpha\sin^2\beta}$ but if I write $\sqrt{\...
4
votes
2answers
120 views

How can I solve $x^\sqrt{y} +y^\sqrt{x} =\dfrac{49}{48}$ and $\sqrt{x}+\sqrt{y} =\dfrac72$?

I have tried Wolfram Alpha and Mathematica to get the solution of the below system, but no result , I have used variable change $z=\sqrt{x}+\sqrt{y}$ for simplification but no result , $$ \left\{ \...
0
votes
1answer
43 views

Proof by induction: Let $a_0=3$ and $a_{n+1}=\sqrt{a_n+7}$ if $n>0$, Prove: $3<a_n<4$

Let $a_0=3$ and $a_{n+1}=\sqrt{a_n+7}$ if $n>0$ Prove: $3<a_n<4$ At first I was quite surprised it's actually true for the base cases: $n=0$, $a_1=\sqrt{3+7}=\sqrt{10}$ $n=1$, $a_2=...
0
votes
3answers
21 views

Find Polynomials with Integer Coefficients with Particular Roots [duplicate]

Is there any simple way to find a polynomial with integer coefficients so that ($x=\sqrt{2} +\sqrt{3}$) is one of its roots? I know one way is to get rid of all the square roots in the equation to be ...
2
votes
4answers
97 views

How to prove AM-GM by induction 3

Let $a_1;a_2;...;a_n\ge 0$. Prove that $$\frac{\sum ^n_{k=1}a_k}{n}\ge \sqrt[n]{\prod ^n_{k=1}a_k}$$ We will prove it's true with $n=k$. Indeed we need to prove it's true with $n=k+1$ WLOG $a_1\le ...
7
votes
4answers
133 views

Prove that inequality $1+\frac{1}{2\sqrt{2}}+…+\frac{1}{n\sqrt{n}}<2\sqrt{2}$

Let $n$ is a natural number. Prove that inequality $$1+\frac{1}{2\sqrt{2}}+\frac{1}{3\sqrt{3}}+...+\frac{1}{n\sqrt{n}}<2\sqrt{2}$$ My try: $$\frac{1}{n\sqrt{n}}=\frac{\sqrt{n}}{n^2}<\frac{\sqrt{...
2
votes
6answers
131 views

How to prove $1 + \frac{1}{\sqrt{2}} +… +\frac{1}{\sqrt{n}} <\sqrt{n}\ .\left(2n-1\right)^{1/4} $

Prove that $$1 + \frac{1}{\sqrt{2}} +... +\frac{1}{\sqrt{n}} <\sqrt{n}\ .\Bigl(2n-1\Bigr)^{\frac{1}{4}} $$ My Approach : I tried by applying Tchebychev's Inequality for two same sets of numbers; ...
3
votes
4answers
134 views

Solve the equation $\sqrt[3]{x^{2}+4}=\sqrt{x-1}+2x-3$

Solve the equation: $$\sqrt[3]{x^{2}+4}=\sqrt{x-1}+2x-3$$ Things I have done so far: $$\sqrt[3]{x^{2}+4}=\sqrt{x-1}+2x-3$$ Deducting 2 from both sides of equation $$\Leftrightarrow (\sqrt[3]{x^2+4}-2)=...
3
votes
2answers
149 views

Let $m$ be the largest real root of the equation $\frac3{x-3} + \frac5{x-5}+\frac{17}{x-17}+\frac{19}{x-19} =x^2 - 11x -4$ find $m$ [closed]

Let $m$ be the largest real root of the equation $$\frac3{x-3} + \frac5{x-5}+\frac{17}{x-17}+\frac{19}{x-19} =x^2 - 11x -4.$$ Find $m$. do we literally add all the fractions or do we do something ...
0
votes
6answers
112 views

Prove that inequality $\frac{2ab}{a+b}+\sqrt{\frac{a^2+b^2}{2}}\ge \sqrt{ab}+\frac{a+b}{2}$

Let $a;b\ge 0$. Prove that inequality $$\frac{2ab}{a+b}+\sqrt{\frac{a^2+b^2}{2}}\ge \sqrt{ab}+\frac{a+b}{2}$$ My try: $LHS-RHS=\frac{2ab}{a+b}-\frac{a+b}{2}+\sqrt{\frac{a^2+b^2}{2}}-\sqrt{ab}\ge 0$ ...
0
votes
0answers
33 views

standardized form for products of surds

As an exercise, I'm writing a program that takes a radical expression* as an input and returns a radical expression equal to it that is in as close to a canonical form as possible. The ideal would be ...
1
vote
2answers
48 views

Expanding log problem

I found this site with online problems and answers. https://courses.lumenlearning.com/waymakercollegealgebra/chapter/expand-and-condense-logarithms/ I've tried several problems and my answer is ...
1
vote
1answer
17 views

Nilpotent Jacobson radical

I have to prove the following: If $R$ is a finite-dimensional algebra over a field $F$, then $J(R)$ is nilpotent. I thought about this, but there are some gaps: Because $R$ is in particular a ...
1
vote
1answer
56 views

Sum of square roots of complex numbers

Is this always true? $z^{1/2}+z^{1/2}=2z^{1/2}$ I said that it isn't in the multiform case but I want to justify this better (not just words).
18
votes
4answers
1k views

How to simplify $\int{\sqrt[4]{1-8{{x}^{2}}+8{{x}^{4}}-4x\sqrt{{{x}^{2}}-1}+8{{x}^{3}}\sqrt{{{x}^{2}}-1}}dx}$?

I have been asked about the following integral: $$\int{\sqrt[4]{1-8{{x}^{2}}+8{{x}^{4}}-4x\sqrt{{{x}^{2}}-1}+8{{x}^{3}}\sqrt{{{x}^{2}}-1}}dx}$$ I think this is a joke of bad taste. I have tried every ...
2
votes
3answers
51 views

A Question About Square Roots And Exponent Laws

Why is it in math, $\sqrt{ab}$=$\sqrt{a}$$\sqrt{b}$? I get why this is the case for any other power instead of $1/2$. For instance, if the power was for, then $(ab)^4$=$(a)^4$$(b)^4$ because on both ...
11
votes
3answers
2k views

Conjugate of real number

I'm slightly confused on the subject of conjugates and how to define them. I know that for a complex number $ a - bi $ the conjugate is $ a + bi $ and similarly for $ 1 + \sqrt 2 $ the conjugate is $ ...
3
votes
3answers
74 views

Find the positive value of $x$ satisfying the given equation

$${\sqrt {x^2- 1\over x}} + {\sqrt{x-1\over x}} = x$$ Tried removing roots. Got a degree $6$ equation which I didn't no how to solve. Also tried substituting $x = \sec(y)$ but couldn't even come ...
0
votes
0answers
92 views

How to solve equations with both logarithms and square roots, like this: $ax+b\log(x)+c\sqrt{x}+d=0$

I have an equation that looks like this: $$ax+b \log(x)+c\sqrt x+d=0$$ I know that an equation without the $\sqrt x$ can be solved using the Lambert's W function (How to solve equations with ...
2
votes
2answers
95 views

Is there a clever way to compute $\int \limits_a^b \sqrt[3]{(b-x)(x-a)^2} dx$

I'm wondering how to compute $\int \limits_a^b \sqrt[3]{(b-x)(x-a)^2} dx$. I tried doing Chebyshe[o]v's substitution to first compute the antiderivative which led me to $\int \frac{w^3}{(w^3+1)^3} dw$ ...
1
vote
3answers
156 views

What is wrong with the reasoning in $(-1)^ \frac{2}{4} = \sqrt[4]{(-1)^2} = \sqrt[4]{1} = 1$ and $(-1)^ \frac{2}{4} = (-1)^ \frac{1}{2} = i$?

$$(-1)^ \frac{2}{4} = \sqrt[4]{(-1)^2} = \sqrt[4]{1} = 1$$ $$(-1)^ \frac{2}{4} = (-1)^ \frac{1}{2} = i$$ Came across an interesting Y11 question that made pose this one to my self. I can't for the ...
0
votes
4answers
40 views

Simultaneous equations with two unknowns

The question I've been given is $$\begin{array}{c|c}t&v\\\hline3&38\\12&200\end{array}$$ Modelling equation is $$v=k\sqrt{t-a}$$ Calculate $a$ and $k$. I tried to solve like: $$38 = k \...
0
votes
3answers
79 views

Find the value of the expression $A=xy^3-x^3y$

Let $$x=\frac{2}{2\cdot \:2^{\frac{1}{3}}+2+4^{\frac{1}{3}}};y=\frac{2}{2\cdot \:2^{\frac{1}{3}}-2+4^{\frac{1}{3}}}$$.Find the value of the expression $$A=xy^3-x^3y$$ I see: $$\left(4^{\frac{1}{3}}-2^...
7
votes
2answers
683 views

Tricky Integral involving radicals

I am trying to evaluate the following definite integral (for $a>0$): $$I=\int_{0}^{1}{{{\left( {{\left( 1-{{x}^{a}} \right)}^{\frac{1}{a}}}-x \right)}^{2}}dx}$$ Neither the substitution $u={{\left(...
2
votes
2answers
194 views

Proof of the irrationality of $\sqrt{2}$.

So I was thinking about how to prove the $\sqrt{2}$ is irrational while on holiday. (Bear in mind that at this point, I only knew that it was indeed irrational, not how to prove it.) Anyway, I came ...
0
votes
0answers
27 views

Square and cubic root of gaussian Integer

I define the integer square root of an integer n as the largest number r with $r^2$ <= n. Is there an analogon for Gaussian integers? With n = c+di I am looking for the number r = a+bi with ...
1
vote
2answers
66 views

How to solve equations in two variables when they're not so appealing?

I was trying to bash IMO 2009/4 using trigonometry and I landed up with the equations : $$3y - 4y^3 = \frac{d+2}{2\sqrt{2} d}$$ $$2y\sqrt{1-y}=\frac{1}{d}$$. From here, I have to write an equation ...
41
votes
3answers
1k views

Does this pattern continue $\lfloor\sqrt{44}\rfloor=6, \lfloor\sqrt{4444}\rfloor=66,\dots$?

By observing the following I have a feeling that the pattern continues. $$\lfloor \sqrt{44} \rfloor=6$$ $$\lfloor \sqrt{4444} \rfloor=66$$ $$\lfloor \sqrt{444444} \rfloor=666$$ $$\lfloor \sqrt{...
-1
votes
1answer
81 views

Derivative of $\sqrt{x}$ Using Symmetric Derivative Formula [closed]

How do you find the derivative of $\sqrt{x}$ using the symmetric derivative formula? $$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x-h)}{2h}. $$ I got stuck on trying to remove the h from the ...