Question:
roor ab is irrational then prove that root a plus root b is irrational
Answer:
Given, √(ab) is an irrational number.
Let √a + √b is rational.
So, √a + √b can be written as p/q form where q ≠ 0
=> √a + √b = p/q
Squaring on both side, we get
=> (√a + √b)2 = (p/q)2
=> a + b + 2√a * √b = p2 /q2
=> a + b + 2√(ab) = p2 /q2
=> 2√(ab) = p2 /q2 - (a + b)
=> √(ab) = {p2 /q2 - (a + b)}/2 ...........1
Since √a + √b is rational, So √a , √b is also rational.
But LHS of equation 1 is irrational.
which contradict our assumption.
Hence, √a + √b is an irrational number.
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