Question

Prove that (6)1/3 is an irrational number​

Answer 1

(6*3+1)/3=19/3=6.33333333.......

its repeating and reoccurring therefore its irrational no.

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Answer 2

Proved below that $6^{\frac{1}{3}}$ is an irrational number.

Step-by-step explanation:

To prove : $6^{\frac{1}{3}}$ is an irrational number ?

Proof :

Assume that $6^{\frac{1}{3}}$ is rational.

Then it can be written as $6^{\frac{1}{3}}=\frac{n}{m}$ for some integers n and m which are co-prime.

Cubing root both side,

$6=\frac{n^3}{m^3}$

So n³ must be divisible by 6 and hence n must be divisible by 6.

Let n = 6p for some integer p.

$6=\frac{(6p)^3}{m^3}$

$6=\frac{6^3p^3}{m^3}$

So m³ and hence m must be divisible by 6.

But n and m where co-prime so they can not have any factors in common so we have a contradiction.

So $6^{\frac{1}{3}}$ must not be rational.

Hence it is irrational.

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