Question

Prove 2 root 3 -5 root 2 is irrational number

Answer 1

Given : 2√3  - 5√2

To Find : Prove that it is irrational

Solution:

Assume that  2√3  - 5√2  is rational

then  2√3  - 5√2 = p/q   where p and q are integers and q ≠ 0

Squaring both sides

=> 12  + 50  - 20√6   = p²/q²

=> 62  - 20√6   = p²/q²

=>  62 - p²/q² = 20√6

=> (62q² - p²)/q² = 20√6

=> (62q² - p²)/20q² =  √6

LHS is rational as p and q are integers

while RHS √6 is rational

Which is not true

Hence our assumption was incorrect

So 2√3  - 5√2  is not rational Hence irrational

2√3  - 5√2  is irrational

QED

Hence proved

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