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I'm having some problem wrapping my head around the sophisticated way to format mathematical problems on this forum, so you will have to excuse me for not expressing my question in the correct format! :)

On to my question:

I was recently presented with this equation, and asked whether or not I thought it was correct:

$((x^2)^2)^3 = x^{64}.$

So I answered that I thought it was correct. I was given the answer that the equation is, in fact, not correct, but was not provided an explanation as to why this is. I eventually managed to deduce, through my brilliant insight into the world of mathematics, that it must be because the actual answer is derived through multiplying the exponents with each other, like so:

$((x^2)^2)^3 = x^{12}$

So the correct answer is that I would take 2 times 2 times 3, which equals 12, instead of 2 to the power of 2 to the power of 3, which equals 64.

So I thought I was supposed to take the power of each exponent with the consecutive exponent, so that I would have the first two exponents, i.e. 2 to the power of 2, which equals 4, and then have 4 to the power of 3, which equals 64. This is apparently not correct, however, as I'm supposed to take the first two exponents times each other, as in 2 times 2, and then take the resulting 4 times 3, which equals 12, so that I end up with $((x^2)^2)^3 = x^{12}.$

My question is why this is? I know that an exponent on an exponent, like $(x^2)^2$, is supposed to be multiplied with each other, so that I would take the exponent of 2 times the other exponent of 2, and get the result being the exponent of 4. In the example above, I would then take the exponent of 4 and multiply it with the exponent 3, which would result in x^12. I'm thinking that we're not dealing with "mere" multiplication though, but with "the power of"! Shouldn't we treat the problem as such, then?

Is there a way for you fine people to explain to a dunce like me why it's logically correct to simply multiply the exponents, rather than take the exponents to the power of the consecutive exponent, and then take the result of that with the next consecutive exponent, if we have a total of three exponents?

TheCatHat
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    Compare to $2^{2^{2^{3}}} \neq ((2^2)^2)^3$ – amWhy May 28 '22 at 18:22
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    maybe you will find better the difference if you see $x^{2^{2^3}}$ and then $((x^2)^2)^3$. Just notice that $x^{2^{2^3}} = x^{2^{8}} = x^256$. Also, $(x^2)^2$ means $(x^2)\times(x^2)$, so it is $x^4$ – L F May 28 '22 at 18:25
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    You need to understand that the brackets play the same role here as everywhere else in algebra, i.e. : calculate the expression in the brackets first, and the result is the same as the one obtained when the whole bracket is replaced by its result. In other words, because $(2^2)^2=2^4$, we have $((2^2)^2)^3=(2^4)^3$ (outer bracket evaluated) $=2^{12}$. –  May 28 '22 at 18:29
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    https://math.stackexchange.com/questions/239825/can-anyone-explain-why-abc-abc-neq-abc-abc – Etemon May 28 '22 at 18:33
  • Thank you very much, amWhy, L F, Stinking Bishop and Etemon! I think I'm starting to get closer to understanding the logic behind the solution! :) – TheCatHat May 28 '22 at 18:56
  • If you ignore the conventions, and just think in terms of using parenthesis or brackets to explicitly define the order of operations, the confusion comes from $$a^{(b \times c)} = \left(a^b\right)^c \neq ~[ ~\text{in general} ~] ~a^{(b^c)}.$$ – user2661923 May 28 '22 at 19:31

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For this it would be best to look at exponential rules which essentially says $\left( x^{m} \right)^n = x^{mn}$ but we can look on how to derive it. Essentially when you are using powers you are using it in order to shorten multiplication. We can write it out and see what happens: $$\left( x^2 \right)^3 = x^2 \times x^2 \times x^2$$ We know that (through exponential laws) that $x^2 \times x^2 \times x^2 = x^{2+2+2} = x^6$. We can expand this and use PEMDAS to larger values such as $\left( \left( x^2 \right)^2 \right)^3$ which would be the same logic as before. We would simplify the outer layer first which would give us $\left( \left( x^2 \right)^2 \right)^3 = \left( x^4 \right)^3 $. We can simplify this further and see that the answer is $x^{12}$.

Hopefully that answered your question!

Soham Garg
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  • Aha, I see! That actually does make sense, now that I think about it. I'm probably going to have to solve more problems like this and apply your answer to truly get an intuitive understanding of the logic, but your explanation with simplifying the problem really helps. Thank you so much! :) – TheCatHat May 28 '22 at 18:51
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    No Problem! Just make sure to take examples for everything and trying simplifying it out next time! – Soham Garg May 28 '22 at 18:53