Power Stack
When you stack powers, how do you evaluate them?
Age
16 to 18
Challenge level
Problem
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.
Student Solutions
$$
3^{(3^3)} = 3^{(27)} = 7625597484987\quad\quad (3^3)^3 = 27^3 = 19683
$$
The difference rapidly grows for larger values:
$$ 4^{(4^4)} = 4^{(256)} \sim 10^{154} \quad\quad (4^4)^4 = 256^4\sim 10^9 $$
However, for $2$ the values are the same
$$ 2^{(2^2)} = 2^{(4)} = 16\quad\quad (2^2)^2 =4^2 =16 $$
The extension of the definitions are naturally either 'powers evaluated from the right' or 'powers evaluated from the left'. The difference for a stack of four powers is gigantic
$$
(((3^3)^3)^3) = (((27)^3)^3) = (19683)^3\sim 10^{12}
$$
$$
(3^{(3^{(3^{(3)})})}) =(3^{(3^{27})}) =(3^{(7.6\times 10^{12})})\sim 10^{3.6\times 10^{12}}
$$
Using a spreadsheet we found that both definition of stacking four numbers leads to the same value when the base is $1.02092370325178$