## Rank the following quantities in order, from smallest to biggest.

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### Rank the following quantities in order, from smallest to biggest.

by Gmat_mission » Fri Feb 11, 2022 8:42 am

00:00

A

B

C

D

E

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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

Source: Magoosh

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### Re: Rank the following quantities in order, from smallest to biggest.

by Ian Stewart » Fri Feb 11, 2022 2:59 pm
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.
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