Question

Prove that √3 is an irrational number. Using it, show that 5 + 2 √3 is not a rational number.​

Answer 1

Answer:

Let √3 is an rational number so it can be written in the form of p/q where p,q have not common factor.p,q are integer such that q is not equal to 0

√3 = p/q

Squaring both sides, 3=p^2/q^2

3q^2=p^2

It means 3 is a factor of q^2 so it's also a factor of q

(By theorem) 3 will also a factor of p

But it contradict our supposition this implies √3 is an irrational number

5+2√3=p/q

2√3=p/q-5

2√3=p-5q/q

√3=p-5q/2q

p is an integer 5q is an integer 2q is an integer so R.H.S will be integer

But it contradict because we already prove that √3 is an irrational number so 5+2√3 is not a rational number. Hence, it Is proved

Hope it may help you