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Question
Prove that `2 + 3sqrt(3)` is an irrational number when it is given that `sqrt(3)` is an irrational number.
Solution
To prove: `2 + 3sqrt(3)` is irrational, let us assume that `2 + 3sqrt(3)` is rational.
`2 + 3sqrt(3) = a/b; b ≠ 0` and a and b are integers.
⇒ `2b + 3sqrt(3)b = a`
⇒ `3sqrt(3)b = a - 2b`
⇒ `sqrt(3) = (a - 2b)/(3b)`
Since a and b are integers so, `a - 2b` will also be an integer.
So,`(a - 2b)/(3b)` will be rational which means `sqrt(3)` is also rational.
But we know `sqrt(3)` is irrational (given).
Thus, a contradiction has risen because of an incorrect assumption.
Thus, `2 + 3sqrt(3)` is irrational.
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