Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Prove it United States

Problem:

Order the numbers $2^{300}$ , $10^{100}$ , and $3^{200}$ from least to greatest, and prove that your ordering is correct.

Solution

Solution:

We note that $2^{300} = (2^{3})^{100} = 8^{100}$ and $3^{200} = (3^{2})^{100} = 9^{100}$ . Since $8^{100} < 9^{100} < 10^{100}$ , the ordering is $2^{300} < 3^{200} < 10^{100}$ .

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