AMC 8 · 2010 · #24

Grade 8 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

exponentspattern-recognition identify-subproblemspattern-recognition ↑ Prerequisites: exponentsmulti-digit-arithmetic

📏 Short solution 💡 3 insights

Problem

Put the three numbers 10⁸, 5¹², and 2²⁴ in order from least to greatest, and pick the matching answer choice.

Pick an answer.

(A)

$2^{24}<10^8<5^{12}$

(B)

$2^{24}<5^{12}<10^8$

(C)

$5^{12}<2^{24}<10^8$

(D)

$10^8<5^{12}<2^{24}$

(E)

$10^8<2^{24}<5^{12}$

AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

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.

1STEP 1

Every exponent is a multiple of 4, so rewrite each number as a 4th power using (a^m)ⁿ = a^mn.

10⁸ = (10²)⁴, 5¹² = (5³)⁴, 2²⁴ = (2⁶)⁴

2STEP 2

Since x↦x⁴ is increasing, ordering the originals is the same as ordering the small bases 10², 5³, 2⁶.

Compare 10² vs. 5³ vs. 2⁶ instead of 10⁸ vs. 5¹² vs. 2²⁴

3STEP 3

Compute the three small powers: 10² = 100, 5³ = 125, 2⁶ = 64 — all easy by hand.

10² = 100, 5³ = 125, 2⁶ = 64

4STEP 4

Order the three results: 64 ＜ 100 ＜ 125.

64 ＜ 100 ＜ 125

5STEP 5

Each small number is the 4th root of its big one, so the same order lifts back: 2²⁴ ＜ 10⁸ ＜ 5¹², which is choice (A).

2⁶ ＜ 10² ＜ 5³ ⟹ 2²⁴ ＜ 10⁸ ＜ 5¹² → (A)

Answer

2²⁴ ＜ 10⁸ ＜ 5¹²

Sanity check with rough magnitudes. 2¹⁰ ≈ 1000, so 2²⁴ = 2²⁰ · 2⁴ ≈ 10⁶ · 16 ≈ 1.6 × 10⁷, which is well under 10⁸. And 5¹² = 5¹² versus 10⁸ = (5 · 2)⁸ = 5⁸ · 2⁸ = 5⁸ · 256, while 5¹² = 5⁸ · 5⁴ = 5⁸ · 625. Since 625 ＞ 256, 5¹² ＞ 10⁸. Both spot checks agree with 2²⁴ ＜ 10⁸ ＜ 5¹².

💡Key takeaway

When 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.

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