SOLUTION: prove cube root 6 is irrational

Question 243988: prove cube root 6 is irrational
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Prove

is irrational.

Assume

is rational. That means

where

is reduced to lowest terms, that is

and

have no common integer factors.

It follows then that:

Since at least one factor of the RHS is even, the entire RHS must be even. Since the RHS is even, the LHS must therefore also be even. Since the product of two odd numbers is always odd, it follows that:

Since

is even, it follows that

So:

Now, since at least one factor of the LHS is even, the LHS must be even. Therefore the RHS must be even and

must be even. But if

is even, then

must be even.

Since the the original assumption leads to the conclusion that both

and

are even, contradicting the part of the original assumption that

and

have no common integer factors, the assumption that

is rational must be false.

Therefore

is irrational. QED.

John

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