AMC 10 · 2022 · #11
Grade 11 algebraPick an answer.
Nobody expands a 2022nd power. So the real question is what these two bases do when you keep multiplying them by themselves, and tool #5 (Look for a Pattern) is the whole engine: if the powers repeat on a short cycle, then f repeats on that same cycle and a huge exponent costs nothing. Tool #9 (Solve an Easier Related Problem) is how the pattern gets found — replace n=2022 with n=1,2,3, which is arithmetic a person can actually do, and see what turns up. Tool #4 (Introduce a Variable) keeps the writing honest: give the two bases the names ω and ω so that facts about them can be stated once and reused, instead of dragging (-1+i√(3))/2 through every line. Tool #15 (Organize Information in More Ways) does the last lift: once the cycle length is known, 2022 has to be re-read not as a size but as a multiple of that cycle length, and the answer falls out of the rewriting.
Name the two bases
The two are conjugates.
Naming the two numbers turns a formula you have to re-copy into a fact you can reuse.
11.N-CN.A.1Introduce A VariableSquare it
Squaring gives the other one.
When a huge exponent blocks the way, compute the smallest powers first and let them tell you what kind of number you are holding.
11.N-CN.A.2Solve An Easier Related ProblemCube it
Cubing gives one.
Multiplying a number by its conjugate wipes out the imaginary part and leaves the squared distance from the origin — here that distance is exactly 1.
11.N-CN.A.3Solve An Easier Related ProblemThe values repeat every three
They repeat every three.
If the base returns to 1 after three steps, the whole list of values is a three-beat loop played over and over.
If the base returns to one after three steps, the whole list of values is a three-beat loop played over and over.
▸ Why?
Once a value repeats, everything that followed it the first time follows it again in the same order.
▸ Why?
Inside a loop of fixed length only the remainder after dividing by that length decides where a step lands.
Reduce 2022 modulo three
The digit sum shows it is a multiple of three.
Once you know the cycle length, a giant exponent carries only one piece of information: its remainder.
4.OA.B.4Organize Information In More WaysCollapse the power
Both become one, so the sum is 2.
Grouping the exponent into blocks of three turns a 2022-fold product into 674 copies of the number 1.
8.EE.A.1Solve An Easier Related ProblemWhen a base returns to 1 after a few multiplications, its powers run in a short loop — so a monstrous exponent like 2022 only ever asks one question: what is the remainder?
- Name the two bases
- Square omega and read the result
- Cube it and land on 1
- The values repeat every three
- Read 2022 as a multiple of 3
- Collapse the power and finish