Competition · AMC preparation · step 4 of 4
AMC 8 · 2010 · #24
Grade 8 arithmeticPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Comparing 9-digit numbers head-on is messy, but the exponents 8, 12, 24 all share the factor 4. So we replace the original problem with the easier related problem of comparing the fourth roots — Tool #9. Taking a fourth root preserves order (the function x ↦ x¹/4 is increasing on positive numbers), so the ordering of the fourth roots is exactly the ordering of the originals. Tool #7 then turns the three-way comparison into easy pairwise subproblems on the small numbers 10², 5³, 2⁶, which any 6th grader can evaluate.
Rewrite each as a 4th power
Every exponent is a multiple of 4, so rewrite each number as a 4th power using (a^m)ⁿ = a^mn.
Sharing a common exponent (4) is the simplification that makes the comparison easy — only the bases 10², 5³, 2⁶ will matter.
8.EE.A.1Solve An Easier Related ProblemCompare the fourth roots
Since x↦x⁴ is increasing, ordering the originals is the same as ordering the small bases 10², 5³, 2⁶.
This is the Tool #9 move: replace a hard problem with an order-equivalent easier one whose numbers fit on one line.
Putting the three big numbers 10⁸, 5¹², 2²⁴ in order is the same as putting the three smaller bases 10², 5³, 2⁶ in that same order.
▸ Why?
Each big number is exactly its smaller base raised to the fourth power: 10⁸ = (10²)⁴, 5¹² = (5³)⁴, and 2²⁴ = (2⁶)⁴.
▸ Why?
A power is its base multiplied together many times, and regrouping the eight equal factors of 10⁸ into four pairs of (10 · 10) gives the same product (10²)⁴; the same regrouping turns 5¹² into (5³)⁴ and 2²⁴ into (2⁶)⁴.
▸ Why?
Raising to the fourth power keeps positive numbers in the same order, so whichever base is larger also has the larger fourth power.
▸ Why?
A fourth power is the base taken as equal groups and multiplied again and again, and equal groups of a larger positive amount always add up to a larger total than the same groups of a smaller amount.
Evaluate the small powers
Compute the three small powers: 10² = 100, 5³ = 125, 2⁶ = 64 — all easy by hand.
Computing 10², 5³, 2⁶ separately is the Tool #7 subproblems move — three tiny independent calculations.
6.EE.A.1Identify SubproblemsOrder the three results
Order the three results: 64 ＜ 100 ＜ 125.
Once everything fits inside three-digit numbers, ordering them is just a quick number-line comparison.
5.NBT.A.2Identify SubproblemsLift the order back up
Each small number is the 4th root of its big one, so the same order lifts back: 2²⁴ ＜ 10⁸ ＜ 5¹², which is choice (A).
The fourth-power function preserves order, so the easier comparison transfers cleanly to the original one.
8.EE.A.1Solve An Easier Related ProblemWhen numbers are too big to compute, look for a shared exponent — taking the same root of all of them keeps the order but shrinks the numbers to something you can handle by hand.
- Rewrite each as a 4th power
- Compare the fourth roots
- Evaluate the small powers
- Order the three results
- Lift the order back up
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