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Question
Prove that `7 + 4sqrt(5)` is an irrational number, given that `sqrt(5)` is an irrational number.
Solution
Let `7 + 4sqrt(5)` is a rational number
∴ `7 + 4sqrt(5) = p/q` ...[where p and q are integers and q ≠ 0]
`7q + 4sqrt(5)q` = p
`sqrt(5) = (p - 7q)/(4q)`
p and q are integers
∴ `(p - 7q)/(4q)` is a rational no. while
`sqrt(5)` is an irrational number
So `sqrt(5) ≠ (p - 7q)/(4q)`
Hence our assumption is wrong
So `7 + 4sqrt(5)` is an irrational number by contradiction fact.
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