No, Manhattan is correct with their explanation. Here's why:
(5x2x2)^(N)
=(5x4)^(N)
=5^(N) x 4^(N) or 5^(N) x 2^(2N) since 4=2^2

5X2X2)^N

= 5^N X 2^2N


Shouldnt this be: 5^N X 4^2N (based on the rule that A^2 X B^2 = AB^2)


Here is your mistake: it's not 5^N*4^2N, it's 5^N*4^N, since (5*2*2)^N = (5*4)^N = 5^N*4^N. Anyway, since 2^2 = 4, we can safely say that 4^N = (2^2)^N = 2^2N

Thank you both for replying, which i guess leads to one more question to help me clarify this.

I know that if you are multiplying 2 numbers with the same base, you add the exponents (X^A * X^B = X^A+B)

And if multiplying 2 numbers with the same exponents you multiply the bases (A^X * B^X = AB^X)

What happens when you have BOTH the same base and exponents?
(i.e. 2^5 * 2 ^5) shouldn't this technically equal 4^10?? (which isnt right)

Nope it shouldn't. (2^5)*(2^5) = 2^(5+5) = 2^10. You don't do the two operations at the same time, you settle for the first one. It's like you described it here:

I know that if you are multiplying 2 numbers with the same base, you add the exponents (X^A * X^B = X^A+B)

The only difference is that you just consider A = B.

Thonk02 wrote:Thank you both for replying, which i guess leads to one more question to help me clarify this.

I know that if you are multiplying 2 numbers with the same base, you add the exponents (X^A * X^B = X^A+B)

And if multiplying 2 numbers with the same exponents you multiply the bases (A^X * B^X = AB^X)

What happens when you have BOTH the same base and exponents?
(i.e. 2^5 * 2 ^5) shouldn't this technically equal 4^10?? (which isnt right)

If they are both you settle for one or the other like Dana said....

See if you multiple the bases you get 4^10.....OR if you add the exponents you get 2^10.....now if you do the math they BOTH yield 1024....