Question

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

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

Answer 1

The correct option is A

irrational number

On expanding ${(1+\sqrt{3})}^2$ using the formula for ${(a+b)}^2$, 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 irrational number.
Since multiplying a rational number with an irrational number results in an irrational number, $2\sqrt{3}$ an irrational number.

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

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