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## 'Power Stack' printed from http://nrich.maths.org/

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Kimberly wants to define $3^{3^3}$ as $(3^3)^3$ but Nermeen thinks that such a stack of powers should be defined as $3^{(3^3)}$ .

Do their definitions lead to the same numerical value? Is the same true if $3$ is replaced with some other number?

How would Kimberly's and Nermeen's definitions most naturally extend to the definition of $3^{3^{3^3}}$? Do their definitions lead to the same numerical value? Is the same true if $3$ is replaced with some other number?

Extension: Try to compute the approximate size of the numbers as powers of 10.

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Both definitions of powers are equally valid, and in mathematics it should be clear from the context as to which to apply: mathematicians often include the brackets to avoid ambiguity. Kimberly's definition of powers is often relevant in mathematics problems whereas Nermeen's definition of powers is often relevant in computer science problems.