Question

Which of the following conditions imply that the real number $x$ is rational
I. $x^{1/2}$ is rational

II. $x^2$ and $x^5$ are rational

III. $x^2$ and $x^4$ are rational

• A. I and II only
• B. I and III only
• C. II and III only
• D. I, II and III

Answer 1

The correct option is C II and III only

We know : Square root of all prime numbers is irrational number and all prime numbers are real numbers , So

If , x is any real number , So $x^{\frac{1}{2}}$ = $\sqrt{x}$ can be a irrational number ,

For example at x = 2 we get $\sqrt{2}$ is an irrational number.

But any real power of any real number always gives a rational number ,

Therefore

$x_2,x_{5}and{x}_4$ will always be rational number at any real value of ' x '

For example at x = 2 we get : $x_2=2^2$ = 4 that is a rational number , $x_4=2^4$ = 16 that is a rational number , $x_5=2^5$ = 32 that is a rational number .