Question

Which of these is true?
$\left(I\right)$ $5\sqrt{3}$ is not a rational number
$\left(II\right)$ $1$ is not the cube of a rational number
$\left(III\right)$ If a is rational and $n$ is an integer greater than $1$, then $a^n$ is rational.

• A. $I$ and $II$
• B. $II$ and $III$
• C. $III$ and $I$
• D. all three

Answer 1

The correct option is C $III$ and $I$

(I) In $5\sqrt{3}$

5 ia s rational number and $\sqrt{3}$ is an Irrational number

As we know, The product of a rational and irrational number is an irrational number.

So, $5\sqrt{3}$ is not a rational number.

Hence, the option (I) is true

(II) 1 is a rational number

and cube of 1 is 1 only, which is a rational number

Hence the option (II) is False

(III) We know that product of two rational number is always a rational number

Hence if a is a rational number and n is greater than one

Then,

a²= a x a is a rational number.

a³ = a2 x a is a rational number,

a⁴ = a³x a is a rational number,

......

......

∴ aⁿ = a^n-1 x a is a rational number.

So, the option (III) is true