I am having trouble understanding the difference between expressions such as (x^6) * (x^4) and (x^6)^4. Can you help me understand the...

In general, there are 2
rules:

The first rule: if we are multiplying 2 power
functions, with matching bases, the exponents are added.

In
this case, we have (x^6) * (x^4). The matching base is x and the exponents are 6 and
4.

(x^6) * (x^4) = x^(6+4) =
x^10

We could also write x^6 as 6 times
x.

x^6 = x*x*x*x*x*x

This is
also the case in which the bases are matching and we'll calculate the sum of
exponents:

x^6 = x^(1+1+1+1+1+1) =
x^6

We'll do the same with x^4 =
x^(1+1+1+1)

The second rule: If we'll raise a power to
another power, the exponents are multiplied.

(x^6)^4 =
x^(6*4) = x^24

Now, let's treat the problem in this
way:

If x^6 =
x^(1+1+1+1+1+1)

(x^6)^4 = x^6*x^6*x^6*x^6 =
x^(6+6+6+6) = x^24 = x^(6*4)