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For a positive integer n, the value of (√3 + 1)^(2n) - (√3 - 1)^(2n) is:

1. an irrational number
2. an odd positive integer
3. an even positive integer
4. a rational number other than a positive integer

Correct answer: an irrational number

Solution

The expression involves the difference of two terms, one of which is irrational and the other is a smaller irrational number, leading to an overall irrational result. The term (√3 + 1)^(2n) grows significantly larger than (√3 - 1)^(2n), but since both terms are irrational, their difference remains irrational.

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