AMC 10 · 2022 · #8
Grade 8 algebraPick an answer.
The product never ends, so there is nothing to compute directly. Tool #4 (Introduce a Variable) is the way in: give the finished value a name P and then hunt for an equation P must satisfy. Tool #5 (Look for a Pattern) supplies the key structural fact — each factor is the cube root of the one before it, so the list is self-similar. Tool #16 (Change Focus) is the move that finishes it: instead of staring at P, look at P³, because cubing strips one cube root off every factor at once and makes a full copy of the original product reappear. That turns an infinite multiplication into a one-line equation.
Name the layers
Give each factor a name.
Stacked radicals look frightening until you name the layers — then every layer is just "cube root of the last one."
8.EE.A.2Look For A PatternName the finished value
Name the whole product too.
You cannot multiply forever, but you can name the answer and then force it to reveal an equation about itself.
6.EE.B.6Introduce A VariableCube the entire product
Cubing lifts every layer by one.
Cubing peels exactly one cube root off every factor at once, which slides the whole list one place to the left.
8.EE.A.1Change Focus Count The ComplementSpot the copy inside
A copy of itself hides inside.
An endless list is unchanged when you shift it one slot, so the product quietly contains a full copy of itself.
An endless list is unchanged when you shift it one slot, so the product quietly contains a copy of itself.
▸ Why?
After the shift the list looks exactly as it did before, so the same value must come out.
▸ Why?
Each factor is the previous one taken to the same fixed root, which is what makes the shift a clean scaling.
Divide one copy out
Dividing leaves just a square.
Cancelling P is legal precisely because a product of numbers all larger than 1 can never be zero.
7.NS.A.2Introduce A VariableTake the positive root
The positive root is root ten.
A squared value pins the size but not the sign, and positivity of the factors settles the sign immediately.
8.EE.A.2Introduce A VariableWhen a product never ends, give it a name and cube it — the whole product reappears inside its own cube, so P³ = 10P and the endless multiplication collapses to P = √(10).
- Name the layers of the product
- Give the finished value a name
- Cube the entire product
- Spot the copy of P inside
- Divide out one copy of P
- Take the positive square root