Question

Prove that 3+2√3 is an irrational no.​

Answer 1

Step-by-step explanation:

We will prove it in a same way as we prove that 3–√3 is an irrational number..

Let us assume that (5−23–√)2(5−23)2 is a rational number.

Then (5−23–√)2=pq(5−23)2=pq where p and q are co-prime.

=>=> 25−2×5×23–√+(23–√)2=pq25−2×5×23+(23)2=pq [by using (a−b)2=a2−2ab+b2(a−b)2=a2−2ab+b2

25−203–√+4×3=pq25−203+4×3=pq

25−203–√+12=pq25−203+12=pq

37−203–√=pq37−203=pq

37+pq=203–√37+pq=203

3720+p20q=3–√3720+p20q=3

Clearly L.H.S. is a sum of two rational number and therefore L.H.S is rational.

So 3–√3 is a rational number.

But we know that 3–√3 is an irrational number.So our assumption is wrong.

Hence (5−23–√)2(5−23)2is an irrational number.