Power of 3
Age
14 to 16
Challenge level
What power of 27 is needed to get the correct power of 3?
Problem
What number needs to replace the ★ as the power of 27, so that
$3^{2016}+9^{1008}+27^{★}=3^{2017}$?
This problem is adapted from the World Mathematics Championships
Student Solutions
Answer: $672$
$9^{1008}=(3^2)^{1008}=3^{2\times 1008}=3^{2016}$
$27=3^3$, so $27^★=(3^3)^★=3^{3\times ★}$
$3^{2016} + 3^{2016} + 3^{3\times ★}= 3^{2017}$
$ = 3^1\times3^{2016}$
$\therefore 3\times$★ $=2016$ also, which means ★ must be $672$.