Question

Prove that root 6 + root 7 is an irrational number

Answer 1

6 + √7 is irrational number

__________ [TO PROVE]

Let us assume that, 6 + √7 is a rational number

=> 6 + √7 = $\dfrac{a}{b}$

Here, a and b are co-prime numbers.

=> √7 = $\dfrac{a}{b}$ - 6

=> √7 = $\dfrac{a\:-\:6b}{b}$

Here..

$\dfrac{a\:-\:6b}{b}$ is rational number.

So, √7 is also a rational number.

But we know that √7 is irrational number.

So, our assumption is wrong.

6 + √7 is irrational number.

___________ [HENCE PROVED]