Rank the following quantities in order, from smallest to biggest.
I. \(2^{60}\)
II. \((2\sqrt2)^{35}\)
III. \(3^{42}\)
(A) I, II, III
(B) I, III, II
(C) II, I, III
(D) II, III, I
(E) III, II, I
Answer: C
Source: Magoosh
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Rank the following quantities in order, from smallest to biggest.
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2 √2 = 2 * 2^(1/2) = 2^(3/2), so we can rewrite item II:
(2 √2)^35 = (2^3/2)^35 = 2^105/2
and since 105/2 < 60, this is less than 2^60.
To compare 2^60 and 3^42, we can notice that 2^60 = (2^3)^20 = 8^20, and 3^42 = (3^2)^21 = 9^21, and since 8^20 has both a smaller base and smaller exponent than 9^21, it is the smaller number (here using the fact that the bases are positive integers), so the answer is II, I, III.
(2 √2)^35 = (2^3/2)^35 = 2^105/2
and since 105/2 < 60, this is less than 2^60.
To compare 2^60 and 3^42, we can notice that 2^60 = (2^3)^20 = 8^20, and 3^42 = (3^2)^21 = 9^21, and since 8^20 has both a smaller base and smaller exponent than 9^21, it is the smaller number (here using the fact that the bases are positive integers), so the answer is II, I, III.
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