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anemone
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Let $x$ be a non-zero number such that $x^4+\dfrac{1}{x^4}$ and $x^5+\dfrac{1}{x^5}$ are both rational numbers. Prove that $x+\dfrac{1}{x}$ is a rational number.
A rational number is any number that can be written as a ratio of two integers, where the denominator is not equal to zero. In other words, it is a number that can be expressed as a fraction.
To prove that $x+\dfrac{1}{x}$ is rational, we can use the fact that any rational number can be written as a fraction. We can rewrite the expression as $\dfrac{x^2+1}{x}$, which is a ratio of two integers and therefore a rational number.
No, $x+\dfrac{1}{x}$ cannot be irrational. As stated before, any rational number can be written as a fraction, and since $x+\dfrac{1}{x}$ can be written as a fraction, it must be rational.
Yes, there are exceptions to this proof. If $x$ is equal to zero, then $x+\dfrac{1}{x}$ is undefined and therefore not rational. Additionally, if $x$ is an imaginary number, then the expression may not be rational.
This proof is relevant in mathematics because it demonstrates the properties of rational numbers and how they can be written as fractions. It also shows the importance of understanding the fundamental concepts of numbers and their relationships in mathematical equations.