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It is similar to the below example.
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Let us assume on contrary that √3+√5 is a rational number.
Then there exists co-prime positive integers pand q such that
=> √3+√5=p/q
=> p/q-√3=√5
=> (p/q-√3)²=(√5)² {squaring on both sides}
=> p²/q² - 2√3/q×p + 3=5
=> p²/q² - 2=2√3/q×p
=> p²-2q²/q²=2p/q×√3 {taking LCM}
=> p²-2q²/2pq=√3
=> √3 {p,q are integers therefore,p²-2q²/2pq is rational}
but this contradicts the fact that√3 is irrational. so our assumption is wrong.
hence,√3+√5 is irrational
Then there exists co-prime positive integers pand q such that
=> √3+√5=p/q
=> p/q-√3=√5
=> (p/q-√3)²=(√5)² {squaring on both sides}
=> p²/q² - 2√3/q×p + 3=5
=> p²/q² - 2=2√3/q×p
=> p²-2q²/q²=2p/q×√3 {taking LCM}
=> p²-2q²/2pq=√3
=> √3 {p,q are integers therefore,p²-2q²/2pq is rational}
but this contradicts the fact that√3 is irrational. so our assumption is wrong.
hence,√3+√5 is irrational
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