2010 AMC 8 Problem 24

Attempt Problem 24 of the 2010 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 8 solutions, or check the answer key.

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24.

What is the correct ordering of the three numbers,

10^8,

5^{12},

and

2^{24}?

2^{24} \lt 10^8 \lt 5^{12}

2^{24} \lt 5^{12} \lt 10^8

5^{12} \lt 2^{24} \lt 10^8

10^8 \lt 5^{12} \lt 2^{24}

10^8 \lt 2^{24} \lt 5^{12}

Answer: A

Concepts:exponent

Difficulty rating: 1480

Small hint:

Compare

2^{24}

,

10^8=(2\cdot5)^8

, and

5^{12}

.

Big hint:

Rewrite the powers so they have comparable eighth-power factors.

Solution:

First, we get

2^{24} = (2^8)(4^8) < (2^8)(5^8) = 10^8.

Next, we get

10^8 = (2^8)(5^8) =

(4^4)(5^8) < (5^4)(5^8) = 5^{12}.

This means

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

Thus, the answer is A .