What is the correct ordering of the three numbers, 10 8 , 5 12 , and 2 24 ?

AMC 8 2010 Problem 24: Algebra Solution

Problem 24 · 2010 AMC 8 Medium

Algebra & Patterns match-bases-or-exponents

What is the correct ordering of the three numbers, 10⁸, 5¹², and 2²⁴?

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Answer: A — 2^24 < 10^8 < 5^12.

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Hint 1 of 2

You can't compare powers with different bases and different exponents. So make one of them match — the exponents 8, 12, 24 all share the factor 4, so force every exponent down to 4.

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Hint 2 of 2

a^mn = (a^m)ⁿ: rewrite each as something-to-the-4th. Once the exponents agree, just compare the bases.

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Approach: rewrite all three with a common exponent of 4

The exponents 8, 12, 24 are all multiples of 4, so pull each into a 4th power: 10⁸ = (10²)⁴ = 100⁴, 5¹² = (5³)⁴ = 125⁴, 2²⁴ = (2⁶)⁴ = 64⁴.
Now the exponent is identical (4), so larger base means larger number: 64 < 100 < 125.
Therefore 2²⁴ < 10⁸ < 5¹².
Why this transfers: to compare exponentials, match either the bases or the exponents — whichever the numbers make easy. A shared exponent factor (here 4) is the giveaway to convert and just eyeball the bases.

Another way — factor out a common eighth power, compare pairwise:

2²⁴ vs 10⁸: write 2²⁴ = 2⁸·4⁸ and 10⁸ = 2⁸·5⁸. The shared 2⁸ cancels and 4 < 5, so 2²⁴ < 10⁸.
10⁸ vs 5¹²: write 10⁸ = 4⁴·5⁸ and 5¹² = 5⁴·5⁸. The shared 5⁸ cancels and 4⁴=256 < 625=5⁴, so 10⁸ < 5¹².
Chaining: 2²⁴ < 10⁸ < 5¹².

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