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प्रश्न
Prove that 3 + 2`sqrt5` is irrational.
उत्तर
If possible let a = 3 + 2`sqrt5` be a rational number.
We can find two co-prime integers a and b such that `3 + 2sqrt5 = a/b,` where b ≠ 0
`(a - 3b)/b`
= `2sqrt5`
= `(a - 3b)/(2b)`
= `sqrt5`
∵ a and b are integers,
∴ `(a - 3b)/(2b)`
= `"Integer - 3 (interger)"/"2 interger"`
= `(a - 3b)/ (2b)` is rational.
= From (1), `sqrt 5` is rational.
= But this contradicts the fact that `sqrt5` is rational.
∴ Our supposition is wrong.
Hence, `3 + 2sqrt5` is irrational.
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