AMC 8 · 2023 · #6

Grade 6 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

place-valuemulti-digit-arithmeticoptimization-counting systematic-enumerationcasework ↑ Prerequisites: place-valuemulti-digit-arithmetic

📏 Short solution 💡 2 insights 📊 Diagram

Problem

The digits $2,0,2,$ and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: The four digits 2, 0, 2, 3 are placed one per box into the expression (base)^(exponent) × (base)^(exponent). Find the largest value the expression can take.

Givens: Digits available: 2, 0, 2, 3 (each used exactly once); Expression shape: a product of two powers — (base)^(exponent) × (base)^(exponent); Exactly one digit per box

Unknowns: The maximum possible value of the expression

Understand

Restated: The four digits 2, 0, 2, 3 are placed one per box into the expression (base)^(exponent) × (base)^(exponent). Find the largest value the expression can take.

Givens: Digits available: 2, 0, 2, 3 (each used exactly once); Expression shape: a product of two powers — (base)^(exponent) × (base)^(exponent); Exactly one digit per box

Plan

Primary tool: #2 Make a Systematic List

Secondary: #3 Eliminate Possibilities

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.

Execute — Answer: C

#3 Eliminate Possibilities 6.EE.A.1 Step 1

To make a product of two powers as large as possible, neither factor should be 0.
If 0 is placed as a base, then 0^2 = 0 or 0^3 = 0, killing the whole product.
So 0 must be in an exponent slot.

$$0^{2} \times \text{(anything)} = 0,\quad 2^{0} = 1$$

💡 Knowing how exponents work (any nonzero number to the 0 power is 1) tells us instantly where the 0 must go.

#2 Make a Systematic List 6.EE.A.1 Step 2

With 0 in an exponent slot, that term equals 1.
The expression collapses to 1 × (base)^(exponent), so we just need to maximize one power built from two of the remaining digits {2, 2, 3}.
The leftover digit pairs with the 0 as a base.

$$x^{0} = 1,\quad 1 \times (\text{base})^{(\text{exp})} = (\text{base})^{(\text{exp})}$$

💡 The zero-exponent rule simplifies the expression to a single power to maximize.

#2 Make a Systematic List 6.EE.A.1 Step 3

List every distinct (base, exponent) pair from {2, 2, 3} systematically: (3, 2), (2, 3), and (2, 2).
Compute each.

$$3^{2}=9,\quad 2^{3}=8,\quad 2^{2}=4$$

💡 Writing every small case down makes the maximum obvious — no case is missed.

#2 Make a Systematic List 3.OA.C.7 Step 4

The largest of 9, 8, 4 is 9, achieved by 3^2.
The leftover 2 pairs with the 0 as base, giving 2^0 = 1.
The maximum expression is 3^2 × 2^0 = 9 × 1 = 9.

$$3^{2} \times 2^{0} = 9 \times 1 = 9$$

💡 Multiplying 9 × 1 is a basic Grade 3 multiplication fact.

[1] #3 6.EE.A.1 To make a product of two powers as large as possible, neither factor should be 0

[2] #2 6.EE.A.1 With 0 in an exponent slot, that term equals 1. The expression collapses to 1 ×

[3] #2 6.EE.A.1 List every distinct (base, exponent) pair from {2, 2, 3} systematically: (3, 2),

[4] #2 3.OA.C.7 The largest of 9, 8, 4 is 9, achieved by 3^2. The leftover 2 pairs with the 0 as

Review

Reasonableness: Check the choices: 16 = 2^4 and 18 = 2 × 9 aren't reachable since we don't have a 4 in our digit set and 3^2 × 2 isn't the allowed form (the second factor must itself be a power). 9 from 3^2 × 2^0 is the largest reachable value, so the answer 9 makes sense.

Alternative: Tool 6 (Guess and Check) — try a few sensible arrangements (3^2 × 2^0, 2^3 × 2^0, 2^2 × 3^0) and pick the biggest. It lands on 9 just as quickly.

CCSS standards used (min grade 6)

6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents (Used the meaning of base^exponent and the special cases 0^n = 0, x^0 = 1 to constrain and evaluate arrangements.)
3.OA.C.7 Fluently multiply and divide within 100 (Used to compute the final product 9 × 1 = 9 as an instant multiplication fact.)

⭐ This AMC 8 problem only needs Grade 6 exponents (any number to the 0 power is 1!) that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

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