Question

Given that under root 3 is irrational,prove that 5+2under root 3 is irrational number.

Answer 1

5 + 2√3 is irrational number

__________ [TO PROVE]

Let us assume that, 5 + 2√3 is a rational number

=> 5 + 2√3 = $\dfrac{a}{b}$

Here, a and b are co-prime numbers.

=> 2√3 = $\dfrac{a}{b}$ - 5

=> 2√3 = $\dfrac{a\:-\:5b}{b}$

=> √3 = $\dfrac{a\:-\:5b}{2b}$

Here..

$\dfrac{a\:-\:5b}{2b}$ is rational number.

So, √3 is also a rational number.

But √3 is irrational number ____ [GIVEN]

So, our assumption is wrong.

5 + 2√3 is irrational number.

___________ [HENCE PROVED]