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Prove that (17)^(1/3) is irrational. You may use the fact that if n^3 is divisible by...

Prove that (17)^(1/3) is irrational. You may use the fact that if n^3 is divisible by 17 then n is divisible by 17

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Answer #1

Let, (17)^(1/3) be rational if possible.

Then, (17)^(1/3) = a/b, where, a & b are integers & gcd (a,b) = 1, i.e. a/b is in the lowest form & b is non-zero

So, 17 = a³/b³

So, 17b³ = a³

So, 17 divides a³

So, 17 divides a

Then, a = 17c, so, a³ = 17³c³

So, 17b³ = a³ = 17³c³

So, b³ = 17²c³

So, 17 divides b³, so, 17 divides b

So, 17 divides a & 17 divides b

So, 17 divides gcd(a,b) = 1

So, 17 divides 1, which is impossible

So, (17)^(1/3) is irrational

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