AMC 10 · 2021 · #1
Grade 6 arithmeticPick an answer.
The expression is one subtraction between two independent chunks, so Tool #7 (Identify Subproblems) splits it: evaluate 2¹⁺²⁺³ on its own, evaluate 2¹+2²+2³ on its own, then subtract. Inside the first chunk the exponent has its own small subproblem, 1+2+3, that must be finished before the power is taken. Tool #5 (Look for a Pattern) supplies the doubling ladder 2, 4, 8, 16, 32, 64, which produces every power of 2 needed without multiplying from scratch. The whole difficulty of the problem is the placement of the exponent, so keeping the two chunks strictly separate is what protects against the trap.
Split the expression into two chunks
Split into two chunks.
An exponent is its own little enclosure: whatever is written up there gets settled before the base is ever raised.
5.OA.A.1Identify SubproblemsEvaluate the single power
Evaluate the single power with the added exponent.
Each step up a power of 2 just doubles the previous value, so counting six doublings from 1 lands on 64.
6.EE.A.1Look For A PatternEvaluate the sum in parentheses
Evaluate the sum in parentheses.
Here the exponents stay on separate powers, so the powers are added, not combined into one big power.
Here the exponents stay on separate powers, so the powers are added rather than combined into one.
▸ Why?
An exponent counts how many times the base is used, and separate powers count separately.
▸ Why?
The value is its terms added together, so each power contributes on its own.
Subtract the two chunk values
Subtracting gives 50.
Once each chunk is a plain number, the original expression is just one two-digit subtraction.
4.NBT.B.4Identify SubproblemsA sum sitting in the exponent is not the same as a sum of powers: finish the exponent first, because adding up there multiplies the powers while adding down here only adds them.
- Split the expression into two chunks
- Evaluate the single power
- Evaluate the sum in parentheses
- Subtract the two chunk values