Prove 3√6 is not a rational number

Question no. 3:- Proved ³√6 is not a rational Number.

Solution:- Let us suppose if possible ³√6 is a rational number and it is in simplest form of P/q as we did in last example √7 is not a rational number.

Then, p and q are integers having no common factor other than 1, and q ≠ 0

Now, p/q = ³√6 => (p/q)³ = 6                                        (i)

 

1³ < 6 < 2³ => 1³ < (p/q) < 2³                                      by using(i)

                  => 1< p/q < 2

If q = 1, then p is an integer such that 1<p<2

But, there is no integer between 1 and 2.

Hence, q ≠ 1 and so q >1

(p/q)³ = 6 => p³/q³ = 6

                 =>P³/q = p³/q = 6q² [multiplying both sides by q²]

Clearly, 6q² is an integer (since q is an integer)

Since p and q have no common factor, p³ and q have no common factor.

Also, q > 1.

So, p³/q is not an integer, while 6q² is an integer.

Consequently, p³/q ≠ 6q².

Thus, we arrive at a contradiction.

Our supposition is wrong.

Hence, ³√6 is not a rational number or ³√6 ≠ Rational number

Remark: Similarly, we can prove that none of the number ³√2, ³√3, ³√4, ³√5, ³√7 etc. are in rational number.

In fact, if m is a positive integer which is not a perfect cube then ³√m is not a rational number.