Question

prove that 2 + root 3 is not a rational number

Answer 1

LET $2 + \sqrt{3}$
=$a \div b$
Square Both sides : -

${(2 + \sqrt{3)} }^{2}$
$4 + 4 \sqrt{3} + 3$
$7 + 4 \sqrt{3}$
$= {a}^{2} \div {b}^{2}$

$4 \sqrt{3} = {a}^{2} \div {b}^{2} - 7$
${a}^{2} \div {b}^{2} - 7$ is a Contradiction.

BUT $4 \sqrt{3}$ is an IRRATIONAL NUMBER.

Therefore
$2 + \sqrt{3}$
is an IRRATIONAL NUMBER.

Hope this helps !