Question

prove that
$\sqrt{3}$
is a irrational number show that 7+2route3 is also on irrational number

Answer 1

Let √3 be rational.
√3=a/b (where a and b are co-primes)
squaring both sides,
3=a^2/b^2
3b^2=a^2......(i)
3 is a factor of a^2,
So 3 is a factor of a.
Put a=3k in equation (i)
3b^2=(3k)^2
3b^2=9k^2
b^2=3k^2
3 is a factor of b^2,
So 3 is a factor of b.
Hence 3 is a factor of both a and b.
=>Our assumption is wrong.
So √3is irrational.