Question

show that 5+2√7 is an irrational number where √7 is
given to be an irrational number ​

Answer 1

Step-by-step explanation:

Let us assume to the contrary that 5 + 2√7 is rational.

5 + 2√7 = a / b. (a and b are coprimes)

(5 + 2√7)² = a² / b²

25 + 20√7 + 28 = a² / b²

20√7 = a² - 53 / b²

√7 = a² - 53 / b² . 20

Since a and b are integers,

a² - 53 / b² . 20 is rational....

But this contradicts the fact that √7 is irrational...

Thus, 5 + 2√7 is an irrational number..