# 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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1answer
59 views

### What is the best way to do X^2= 618

What would be the best and easiest way to do X^2 = 618 (Without a calculator) Wouldn't it just be considered irrational (Solving for x)
1answer
21 views

### Ratio of two sums with inverse radicals

Saw this challenge problem: $$\frac{\sum_{n=0}^{n=\infty} \frac{1}{\sqrt{3n+1}} - \frac{1}{\sqrt{3n+2}}}{\sum_{n=0}^{n=\infty} \frac{1}{\sqrt{6n+1}} - \frac{1}{\sqrt{6n+5}}} = 2 - \sqrt2$$ How to ...
4answers
163 views

### Proving that $\sqrt 7 -\sqrt 2$ is irrational.

I understand proving that $\sqrt{7}-\sqrt {2}$ is irrational, but how does the answer change if its cube root of $7$ instead of square root? the way I solve $\sqrt{7}-\sqrt {2}$ is by assuming its ...
1answer
54 views

### If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.

If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$. My try It is easy to see that if we raise the first equation ...
1answer
18 views

### What is the proper convention regarding the order of operations of a fractional exponent and/or the simplification of it?

Specifically, consider the example $\sqrt{x^2}$. The answer of course would be $\sqrt{|x|}$ since the x is squared first. However if converted to the exponential fraction of $x^{2/4}$, you lose the ...
0answers
53 views

### Why does this nice method work for expressing accurate trigonometric values in the form $\sqrt{\frac{2\pm\sqrt{2\pm\sqrt{2\cdots\pm\sqrt{2}}}}{2}}$?

I am amazed by the nice work of Mr. Daahal on Breaking-Classical-Rules-in-Trigonometry-Exact-Trigonometric-Values. He provided the algorithm without proof. If someone can provide insight on why ...
0answers
37 views

### How to find out if sum of square roots is a perfect(-ish) square: $\sum_{i=0}^m \sqrt{x_i} = \bigl(\sum_{j=0}^n \sqrt{y_j}\bigr)^2$?

I'm writing a Python library that deals with symbolic computations of square roots (since approximated ones cause a lot of problems and at least for now correctness is more valuable than performance). ...
2answers
43 views

### Converting a radical to a fractional exponent

I want to understand how to convert a radical to a fractional exponent. Given the following equation: $\sqrt{(x)^6\cdot x^9}=\sqrt{x^{24}\cdot x^9}=\sqrt{x^{33}}=x^{\frac{33}3}=x^{11}$ How ...
5answers
61 views

### Solving $\lim_{x\to{a}}\frac{x^2-\sqrt{a^3x}}{\sqrt{ax}-a}$ without L'Hopital/derivates [closed]

$$\lim_{x\to{a}}\frac{x^2-\sqrt{a^3x}}{\sqrt{ax}-a}$$ It should be $3a$, but I can't find the way to solve it without L'Hopital.
0answers
41 views

5answers
107 views

1answer
85 views

### Does $2x^5 - x^4 - 22x^3 - 23x^2 + 22x +24 = 0$ have exact solutions in radicals?

Does $2x^5 - x^4 - 22x^3 - 23x^2 + 22x +24 = 0$ have exact solutions in radicals? A mysterious commenter said on Youtube this was the "easiest quintic equation of my life," and I'm ...
3answers
50 views

1answer
29 views

### Please help me clear confusion over principal roots and identities for n-th radicals

From my old high school math textbook: If ${a{\geq }0}$ and $n\in \mathbb{N} ^{\ast }$, then ${\sqrt[{n}] {a}}$ is the non-negative solution of ${{x}^{n}}=a$. It then goes on to infer a number of ...
0answers
127 views

2answers
74 views

### Sequence involving $\sqrt[n]{n}$ and $\text{log}$

I know that the sequence $\sqrt[n]{n}$ converges to 1 and that $\text{log}(\sqrt[n]{n})$ thus converges to 0 as $n\to\infty$ since the logarithmic function is continuous. But how can I calculate the ...
1answer
46 views

### How to calculate fractional roots on a computer? [closed]

I worked out a closed form formula for a function that is defined for positive integers $n$: When $n$ is even: $x = 1 - (\alpha - 1)^n$ When $n$ is odd: $x = (\alpha - 1)^n + 1$ When this is ...
1answer
40 views

### Surd Manipulations.

If a=$-\sqrt {99}+\sqrt {999}+\sqrt {9999}$ b=$-\sqrt {99}-\sqrt {999}+\sqrt {9999}$ c=$-\sqrt {99}+\sqrt {999}-\sqrt {9999}$ Then $\displaystyle\sum_\limits{cyc} \frac{a^4}{(a-b)(a-c)}$ equals?? I ...
1answer
38 views

2answers
61 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 ...
1answer
37 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 \$\...