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√p + √q is irrational number.
Step-by-step explanation:
Let √p + √q be rational.
=> √p + √q = (a/b)
Squaring both sides, we will get
=> (√p + √q)^2 = (a/b)^2
=> p + q + 2√pq = a²/b²
=> 2√pq = (a²/b²) - p - q
=> 2√pq = (a² - p - q)/b²
=> 2√p = (a² - p - q)/b²√q
=> √p = (a² - p - q)/2b²√q
which is a contradiction as the right hand side is rational number, while √p is irrational.
Therefore, √p + √q is irrational.
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