Proving 0 = 6 with x^a=x^b=>a=b

[vissh]

Helllo :D
One of my buddies asked me today to prove 0 = 6.
Then, my other buddy gave this as solution -
Bcoz we knw anything raised to the power 0 is one=>
bcoz
1=1
1^0=1^6
therefore 0=6(as x^a=x^b=>a=b)

Hehe funny . But as much as i remeber i too use x^a=x^b=>a=b . But now seeing this,I think that there must be some restriction on value of x.
What you think :)

 

[Hootenanny]

(as x^a=x^b=>a=b)

Err, no.

$$x^a = x^b \Rightarrow x = x^{b/a} \Rightarrow 1 = x^{b/a - 1}$$

 

[vissh]

Thanks for replying :)

Err, no.

$$x^a = x^b \Rightarrow x = x^{b/a} \Rightarrow 1 = x^{b/a - 1}$$,

hmm as u raised power 1/a both sides , a is not equal to 0.
Then,u divided by x => x is not equal to 0.
Hmmm i got only these restrictions and thus, x could be 1 and
thus , 1^2=1^6
therefore 2=6(as x^a=x^b=>a=b) [replaced 0 by 2 as i got a not equal 0 ]
hehe
So there are more restricitons i guess. can u pls put some light on them :)
Thanks in advance ^.^

 

[Hootenanny]

Thanks for replying :)

hmm as u raised power 1/a both sides , a is not equal to 0.
Then,u divided by x => x is not equal to 0.

Hmmm i got only these restrictions and thus, x could be 1 and
thus , 1^2=1^6
therefore 2=6(as x^a=x^b=>a=b) [replaced 0 by 2 as i got a not equal 0 ]
hehe
So there are more restricitons i guess. can u pls put some light on them :)
Thanks in advance ^.^

So, for x=0 the relation doesn't hold. In a sense, this is obvious since 0 raised to any power is always zero. Therefore, x^a=x^b=>a=b doesn't hold for x=0. The other number which it doesn't hold for is x=1 .

 

[vissh]

So, its common sense that finally comes into play :)

 

[Hootenanny]

So, its common sense that finally comes into play :)

Yes. Although, you could turn your example into a formal proof by contradiction that the relation x^a=x^b => a = b doesn't hold for x=0 or x=1.