2 books I recommend: how to prove it, and the book of proof, get a real analysis book, and if possible take a course. it took me around 5 years to be able to prove stuff... you need relaxation, fluid thinking, and a breadth of knowledge of facts you can use to prove stuff
we can try your example
we want to show that (a^b)^c = a^bc
let's work on the LHS
by the definition of an exponent, we know that a^b is just a * a * ... * a b times, so we can rewrite it as:
(a_0 * a_1 * ... * a_b)^c
by the same definition, we can multiply the quantity inside the parenthesis by itself c times:
side note; there are some subtleties here for the rationals and serious difficulties with this method for irrationals. I agree this is a good start for someone thinking about how to prove some simple math ’rules’ though, so please don’t take this as discouragement; rather encouragement to keep opening up minds when discussing math!
we can try your example
we want to show that (a^b)^c = a^bc
let's work on the LHS
by the definition of an exponent, we know that a^b is just a * a * ... * a b times, so we can rewrite it as:
(a_0 * a_1 * ... * a_b)^c
by the same definition, we can multiply the quantity inside the parenthesis by itself c times:
(a_0 * a_1...a_b)_0 * (a_0 * a_1 * ... * a_b)_1 * ... * (a_0 * a_1 * ... * a_b)_c
now, use the fact that a^m * a^n = a^(m+n) to consolidate the parenthesis, since each factor has an exponent of 1 we can use simple counting:
(a^b)_0 * (a^b)_1 * ... * (a^b)_c
Repeat the previous step c times, we end up with
a^(b_0 + b_1 + ... + b_c)
which of course is just
a^(bc)
therefore (a^b)^c = a^(bc)