Question

if x is a non zero rational number such that both xtothepower of 5 and 20x+19/x are rational numbers. prove that x is a rational number​

Answer 1

Since $x^5$ is rational, so are $(20x)^5=20^5\,x^5$ and $\left(\dfrac{19}{x}\right)^5=\dfrac{19^5}{x^5}.$

Hence $(20x)^5+\left(\dfrac{19}{x}\right)^5$ will also be rational, as the sum of two or more rational numbers is always rational.

Let,

• $a=20x$

• $b=\dfrac{19}{x}$

so that $a+b=20x+\dfrac{19}{x}$ is rational as per the question. And also, $a^5+b^5.$

We see that,

$\longrightarrow a^5+b^5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)$

Hence we get $a^4-a^3b+a^2b^2-ab^3+b^4$ as rational too.

Hence, each term in $a^4-a^3b+a^2b^2-ab^3+b^4$ should also be rational. Thus we can get $a^4$ rational.

$\longrightarrow a^4=(20x)^4=20^4\,x^4$

So we can obtain that $a=20x$ is rational, as the following,

$\longrightarrow a=\dfrac{a^5}{a^4}$

Since both $a^5$ and $a^4$ are rational, so will be $a,$ as the quotient of two non - zero rational numbers is always rational.

This implies $x$ is a rational number since 20 is rational.

Hence Proved!