Competition · AMC preparation · step 4 of 4
AMC 8 · 2023 · #6
Grade 6 algebra
Pick an answer.
AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
There are only four digits to drop into four slots, so the candidate arrangements are very few — Tool 2 (Systematic List) lets us enumerate the relevant ones. First, Tool 3 (Eliminate) trims the search massively: if 0 sits in a base slot the whole product is 0, so 0 must go in an exponent slot. Then a short list of cases settles it.
Avoid making a factor zero
Keep the product big: 0 as a base makes that factor 0, so 0 must sit in an exponent slot.
Knowing how exponents work (any nonzero number to the 0 power is 1) tells us instantly where the 0 must go.
To reach the largest possible value, the digit 0 must be placed in an exponent box, not a base box.
▸ Why?
If 0 sits in a base box, its power equals 0, and a single factor of 0 pulls the whole product down to 0 — the smallest value, never the largest.
▸ Why?
A base of 0 raised to a whole-number power just means 0 multiplied by itself that many times, and repeatedly multiplying 0 always gives 0, so the power equals 0.
▸ Why?
The expression then multiplies that 0 by the other power, and zero groups of any number is nothing at all, so the product is 0.
▸ Why?
If 0 sits in an exponent box instead, that box becomes a factor of 1 (a nonzero digit to the 0 power is 1), and multiplying the other power by 1 leaves it untouched, so the value stays as large as it can be.
Use the zero exponent rule
A base to the 0 power is 1, so the product reduces to one power made from the leftover 2, 2, 3.
The zero-exponent rule simplifies the expression to a single power to maximize.
6.EE.A.1Make A Systematic ListList the base and exponent pairs
List every power from 2, 2, 3: 3²=9, 2³=8, 2²=4.
Writing every small case down makes the maximum obvious — no case is missed.
6.EE.A.1Make A Systematic ListPick the largest value
Biggest is 3², so pair the leftover 2 with 0 as 2⁰=1: 3² × 2⁰ = 9.
Multiplying 9 × 1 is a basic Grade 3 multiplication fact.
3.OA.C.7Make A Systematic ListThis AMC 8 problem only needs Grade 6 exponents (any number to the 0 power is 1!) that you already know!
- Avoid making a factor zero
- Use the zero exponent rule
- List the base and exponent pairs
- Pick the largest value
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