Question

Show that 1/3-√6 is an irrational numbers​

Answer 1

To prove : 6^{\frac{1}{3}}6

3

1

is an irrational number ?

Proof :

Assume that 6^{\frac{1}{3}}6

3

1

is rational.

Then it can be written as 6^{\frac{1}{3}}=\frac{n}{m}6

3

1

=

m

n

for some integers n and m which are co-prime.

Cubing root both side,

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

m

3

n

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=

m

3

(6p)

3

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

m

3

6

3

p

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}}6

3

1

must not be rational.

Hence it is irrational.

Answer 2

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