Competition · AMC preparation · step 4 of 4
AMC 8 · 2013 · #15
Grade 6 algebraarithmeticPick an answer.
AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The three equations have no shared variables, so Tool #7 (Identify Subproblems) is the obvious move — solve each equation by itself, then multiply at the end. Inside each subproblem, isolate the power of the base on one side and ask "which exponent of this base gives that number?" Listing b¹, b², b³, … until you hit the target is Tool #6 (Guess and Check) on a tiny ladder of values.
Solve for p
Subtract 3⁴ = 81 from 90, leaving 3^p = 9 = 3², so p = 2.
Recognizing 9 as 3² is the Grade 6 "whole-number exponents" skill.
The first equation 3^p + 3⁴ = 90 pins the exponent p down to a single whole number.
▸ Why?
Once the known term 3⁴ is removed from the total 90, what is left must itself be a power of 3, and matching that remainder to the powers of 3 names p.
▸ Why?
Subtracting 3⁴ = 81 from both sides of 3^p + 81 = 90 leaves 3^p = 9 standing alone, with the equation still balanced.
▸ Why?
Taking the same amount off both sides undoes the added 81 without changing which numbers are equal, because subtraction reverses addition.
▸ Why?
The known term 3⁴ is four 3's multiplied, 3 × 3 × 3 × 3 = 81, so the notation itself fixes it as a single number to take away.
▸ Why?
The leftover 9 equals 3², and walking up the powers of 3 (3, 9, 27, …) shows 9 appears at exactly one exponent, so p must be that exponent.
▸ Why?
3² means two 3's multiplied, 3 × 3 = 9; each step up the ladder multiplies by another 3 and lands on a strictly larger number, so 9 is reached once and only at exponent 2.
Solve for r
Subtract 44 from 76 to get 2^r = 32 = 2⁵, so r = 5.
Walking up the powers of 2 (2, 4, 8, 16, 32) until you land on 32 is Tool #6 in miniature.
6.EE.A.1Guess And CheckSolve for s
Subtract 5³ = 125 from 1421 to get 6^s = 1296; since 6⁴ = 1296, s = 4.
1296 isn't an instantly recognizable number, but four short multiplications by 6 pin it down — exactly the Tool #6 pattern.
6.EE.A.1Guess And CheckMultiply the three exponents
Multiply the three exponents: 2 × 5 × 4 = 40, choice (B).
After the subproblems are finished, the closing move is just a Grade 3 multiplication of whole numbers.
3.OA.A.1Identify SubproblemsThree equations, one trick: peel off the constant, then ask "which power of this base gives that number?" — all Grade 6 exponent work.
- Solve for p
- Solve for r
- Solve for s
- Multiply the three exponents
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