Which definitions or properties would you use to simplify the expression explain

Exponent multiplication and addition will simplify this expression.
Exponent Addition: a^b * a^c =  a^(b + c)
Exponent Multiplication: (a^ d) ^ e = a ^ (d * e)

Multiplication of 4^8 * 4^(-4) is the same as adding exponents together: 
4^(8 - 4)

Which makes the expression simplify to (4^4). This expression is now to another exponent which is equivalent to multiplying exponents together:

(4^4)^-2  = 4^ (4 * -2) = 4 ^ -8

The expression has been reduced to its simplest form.