Problem Solving

I have a query regarding one of the exponent properties/rules.

(3^2)^4 = 3^8 => 8 is a result of 2 * 4. This is one of the exponent rules.

How to differentiate between these. How to apply the rule in the below case and which one is right.

(3^2)^(2^2) = (3^2)^4 = 3^8 => The value 8 comes as a result of multiplying 2 and 4.

(3^2)^(2^2) = (3^2)^4 = 3^16 => The value 16 comes as a result of 2 raised to the power of 4.

I am sure only one of these can be right. But I am missing something here. Would be glad if someone could elaborate more on this.

Anurag@Gurome wrote:

sam2304 wrote:(3^2)^(2^2) = (3^2)^4 = 3^16 => The value 16 comes as a result of 2 raised to the power of 4.

This is not.

Here is why.

(3^2) = (3*3)
(3^2)^4 = (3^2)*(3^2)*(3^2)*(3^2) = (3*3)*(3*3)*(3*3)*(3*3) = (3*3*3*3*3*3*3*3) = 3^8

Let's take another example
3^2^2^3 = 3^12 [12 = 2*2*3].
3^2^2^3 = 3^256

What I did in the second case was 3^2^2^3 = 3^2^(8) = 3^(256) and this is wrong because I exponentially raised the value of 2 by 3 and 2 by 8 in the next step instead of multiplying everything. So when it involves power raised to powers, we should just multiply all the powers leaving the base alone and shouldn't raise the powers exponentially, right ?

what I do in general is - follow the brackets

first solve whatever it is inside any bracket and that the exponent of that number. (Since the exponent rule always confuses me)

If no brackets in exam then you have to go with the way it is written in the exam. if it is a ladder rising up, start solving at the topmost extreme.

Experts?