Question

w
5) - Prove that 2*V3+1/V7
is an Irrational.
​

Answer 1

Answer :

We know already that √3 , √7 and √21 are irrational numbers.

So now let use assume that

2√3 + 1/√7 is a rational number.

Then,

2√3 + 1/√7 = p/q

{ where p and q are co - primes and q is not equal to zero }

2√3 + 1/√7 = p/q

=> ( 2√21 + 1) / √7 = p/q

Now squaring both sides, we get

=> {( 2√21 + 1) / √7 }^2 = { p/q }^2

=> (84 + 4√21 + 1) / 7 = p^2 / q^2

=> ( 85 + 4√21) = (7•p^2) / q^2

=> 4√21 = { ( 7•p^2 ) / q^2 } - 85

=> 4√21 = ( 7•p^2 - 85•q^2 ) / q^2

=> √21 = ( 7•p^2 - 85•q^2 ) / 4•q^2

Here, √21 is an irrational number.

But ( 7•p^2 - 85•q^2) / 4•q^2 is a rational number.

So L.H.S. is not equal to R.H.S.

This is a contradiction to our assumption that ( 2√3 + 1/√7 ) is a rational number. Thus our assumption is wrong.

Hence ( 2√3 + 1/√7 ) is an irrational number.

NOTE : Here it is directly taken that √3,√7 and √21 is irrational number.

But it can also be proved along with this.