Question

prove that under root 3 + 5 under root 2 is an irrational number

Answer 1

Let us assume that 5+2√3 is rational

5+2√3 = p/q ( where p and q are co prime)

2√3 = p/q-5

2√3 = p-5q/q

√3 = p-5q/2q

now p , 5 , 2 and q are integers 

∴ p-5q/2q is rational

∴ √3 is rational

but we
know that √3 is irrational . This is a contradiction which has arisen due to our wrong assumption.

∴ 5+2√3 is irrational

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Answer 2

If possible, let us assume 3+5√2 as a rational number

Then,(3+5√2)-3             
      ⇒5√2 is rational         ∵[difference of two rational is always a rational]

Then,( 5√2 )×(1/5)
         ⇒√2 is rational         ∵[product of two rationals is always a rational]

But,we know that √2 is an irrational number.

Thus,we arrive to the contradiction that √2 is a rational.
This contradiction arises by assuming 3+5√2 as rational.

Thus,3+5√2 is an irrational number.