Question

if a is irrational then a^2 is
A) rational
B) irrational
C) a perfect square
D) sometimes rational sometimes irrational ​

Answer 1

If $a$ is irrational, then $a^{2}$ sometimes rational and sometimes irrational.

Option D) sometimes rational and sometimes irrational

Step-by-step explanation:

We take two examples to confirm our answer.

Example 1.

Let us take, $a=\sqrt{2}$. Then,

$\quad a^{2}=(\sqrt{2})^{2}$

$\Rightarrow a^{2}=2$, which is a rational number.

So, $\boxed{a\in\mathbb{R-Q}\implies a^{2}\in\mathbb{Q}}$.

Example 2.

Let us take, $a=2+\sqrt{3}$. Then,

$\quad a^{2}=(2+\sqrt{3})^{2}$

$\Rightarrow a^{2}=4+4\sqrt{3}+3$

$\Rightarrow a^{2}=7+4\sqrt{3}$, which is an irrational number.

So, $\boxed{a\in\mathbb{R-Q}\implies a^{2}\in\mathbb{R-Q}}$.