# Questions tagged [rational-numbers]

Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.

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### Tensor product of finite group with group of rational number.

The tensor product of the finitely generated abelian group G with a rational number is zero if and only if G is finite. is it true for any finite group? For example, we take the tensor product of the ...
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### Prime and rational numbers

I came across the following question while studying that is stumping me. Can anyone please help me solve it? Let "$a$" be a prime number greater than $10,000$ and let $x=\sqrt{a}$. Which of the ...
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### defining the value of real numbers raised to rational exponents [closed]

if $b^{1/n}$ where $b$ is a real number and $n$ even is defined as the positive real solution to the equation $x^n=b,$ how did they suddenly decide to define the value of $b^{m/n}$ where $m$ and $n$ ...
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### For $x^3+px+q=0$, one of the solutions is $\sqrt 3-1$. Find the value of rational number $p$ and $q$, and other two solutions of the equation.

I cannot find any error from my answer. What’s wrong?
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### Proving that there is no rational number solution of the equation $x^2-3x+1=0$, extrapolating to a more general problem.

I would like to know whether I have correctly proved the following statement and have correctly extrapolated out a general situation. We're asked two things: a) Prove there is no rational number ...
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### Why does the law of distributivity use the distributive property to prove it exists?

I was looking at the Law of Distributivity and disagreed with how it was proven. It proved that $$\frac ab\bigg(\frac cd + \frac ef\bigg) = \frac ab\cdot \frac cd + \frac ab\cdot \frac ef.$$ It ...
### Prove or refute that $\sum_{n=1}^\infty\frac{1}{n^2+A\varphi(n)+B}$ will be irrational for some choice of integers $A,B\geq 1$
We denote the Euler's totient function as $\varphi(n)$ for integers $n\geq 1$, that is a multiplicative function that counts the number of integers $1\leq k\leq n$ up to the given integer n that ...
### find $f(99)$ where $2f(x-\frac{1}{x}) + f(\frac{1}{x}-x) = 3(x+\frac{1}{x})^2.$
Suppose $f(x)$ is defined for all positive numbers $x$, and $2f(x-\frac{1}{x}) + f(\frac{1}{x}-x) = 3(x+\frac{1}{x})^2.$ Find $f(99)$. Plugging in $x=1$, I was able to get \$3f(0) = 3(1+1)^2 = 3\cdot ...