Prove a^(b^2) - a^(c^2) can be divided by a^b - a^c

[Berrnie]

If a,b,c are positive integers and b>c and a>1. Prove that a^b2- a^c2 can be divided by a^b- a^c

I tried to do something with binomial expansion formula but I didn’t come to anything helpful. I tried to make b dependent on c but it left me with more variables. So i am kinda lost and have no idea how to prove it ?

 

[Deleted member 4993]

If a,b,c are positive integers and b>c and a>1. Prove that a^b2- a^c2 can be divided by a^b- a^c

I tried to do something with binomial expansion formula but I didn’t come to anything helpful. I tried to make b dependent on c but it left me with more variables. So i am kinda lost and have no idea how to prove it ?

Rewrite the question using a^b = x and a^c=y

Please show us what you have tried and exactly where you are stuck.

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[Berrnie]

Thank you for the tip <3
so a^b=x and a^c=y
so now we‘be got x^b-y^c and we have to prove it can be divided by x - y
I made b=c+k when k belongs to positive integers cause b>c and b,c belongs to positive integers
x^c*x^k - y^c
i know we have to get to x^c - y^c so we can use formula xⁿ-yⁿ=(x-y)(x^n-1+x^n-2y+…+y^n-1) and prove thesis
I know we have to somehow put x^c - y^c outside the brackets but I don’t know how
cause If you try to put this x^k before the bracket you got x^k(x^c -(y^c/x^k))
any idea what should I do to get x^c - y^c?