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'Weekly Challenge 17: Power Stack' printed from http://nrich.maths.org/
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.
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.