Challenge Level

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.

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.

Did you know ... ?

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.

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.