Weekly Challenge 17: Power Stack

7039 /www/nrich/html/content/id/7039/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> <ul id="buttonBar"> <li> <a href="http://nrich.maths.org/6401&part=">Warm-up problem</a> </li> <li> <a href="http://nrich.maths.org/324&part=">Try this next</a> </li> <li> <a href="https://nrich.maths.org/discus/messages/27/27.html?1274115048">Ask NRICH </a> </li> <li> <a href="https://nrich.maths.org/discus/messages/27/150355.html">Ask NRICH</a> </li> <li> <a href="http://en.wikipedia.org/wiki/Graham%27s_number">Read all about it</a> </li> <li> <a href="http://nrich.maths.org/7291&part=solution">Last week's solution</a> </li> </ul> <div> 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. </div> <div class="framework"> 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. </div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> $$ 3^{(3^3)} = 3^{(27)} = 7625597484987\quad\quad (3^3)^3 = 27^3 = 19683 $$ 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 $1.02092370325178$ </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> $$ 3^{(3^3)} = 3^{(9)} = 19683\quad\quad (3^3)^3 = 9^3 = 729 $$ 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 $1.02092370325178$ </mdoxml> 2 3 0 0 0 0 1 Weekly Challenge 17: Power Stack A weekly challenge: these are shorter problems aimed at Post-16 students or enthusiastic younger students. Collections Weekly Challenge Admin Stage 5 - Core Mapping Stage 5 Core Mapping Document Indices and Surds Secondary Mapping Document DisplayCabinet