Show that <msqrt> a b c </msqrt> is irrational if a , b , c , and <ms

Makayla Boyd

Makayla Boyd

Answered question

2022-06-24

Show that a b c is irrational if a , b , c, and a + b + c are irrational.

Answer & Explanation

Abigail Palmer

Abigail Palmer

Beginner2022-06-25Added 30 answers

Note the requirement that a , b ,, and c (not just a , b , and c ) should be irrational.
For a correct counterexample, take
a = b = c = 4 3 .
These are irrational (this can be proven similarly to the irrationality of the square root of 2). Then,
a + b + c = 2 3 + 2 3 + 2 3 = 3 2 3 ,
which is also irrational -- because 2 3 is irrational (which can again be proven similarly to irrationality of the square root of 2).
However,
a b c = 4 = 2 ,
which is rational.

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