Prove that √3+√4 is an Irrational number.

Dear Student,

Let us assume that
(√3 + √4) is a rational number.

√3 + √4 = r , where r is a rational number.

Squaring both sides, we get

[√3 + √4 ]² = r²

 ⇒3 + 2√12 + 4 = r²    [ ∵ (a+b)2 = a2+b2+2ab]

 ⇒7 + 2√12 = r²

 ⇒2√12 = r² - 6

 ⇒√12 = [ r² - 6] / 2

R.H.S is purely rational, whereas, L.H.S is irrational.
This is a contradiction.
This means that our assumption was wrong.
Hence, (√3 + √4) is irrational.

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