Question

The number ${(1+\sqrt{3})}^2$ is a/an ____________.

• A. natural number
• B. rational number
• C. irrational number
• D. integer

Answer 1

The correct option is C

irrational number

On expanding ${(1+\sqrt{3})}^2$ using the formula ${(a+b)}^2$, we find that

${(1+\sqrt{3})}^2={\left(1\right)}^2+{\left(\sqrt{3}\right)}^2+2\left(1\right)\left(\sqrt{3}\right)=4+2\sqrt{3}$

$\sqrt{3}$ is an irratioanl number.
Since, multiplying rational with irrational results irrational .
$\therefore 2\sqrt{3}$ an irrational number.

Addition of a rational and irrational number results an irrational number.
So, $4+2\sqrt{3}$ is irrational

Therefore the equation ${(1+\sqrt{3})}^2$ is also irrational.