# Which number is irrational?   A) `sqrt(16)`   B) `sqrt(18)`   C) `sqrt(25)`   D) `sqrt(36)`

degeneratecircle | High School Teacher | (Level 2) Associate Educator

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Surely there were supposed to be square root symbols before the numbers (I edited them in).

In that case, it can be proved that the square root of any non perfect square is irrational. Since 16, 25, and 36 are perfect squares and 18 isn't, only `sqrt(18)` is irrational. This is actually not too hard to prove.

Suppose `sqrt(18)=a/b` for some integers `a,b.` Then squaring both sides and multiplying through by `b` gives `18b^2=a^2.` If we factor 18 into primes, we have

`2*3^2b^2=a^2.` Considering the prime factorizations of `a^2` and `b^2,` we see that the right side must have an even number of factors of `2,` while the left side has an odd number. Since this is impossible, this proves that there can be no integers `a,b` for which `sqrt(18)=a/b.`

pramodpandey | College Teacher | (Level 3) Valedictorian

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