Manhattan GMAT Challenge Problem of the Week - 21 June 2011

by Manhattan Prep, Jun 21, 2011

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Question

Which of the following quantities is the largest?

(A) [pmath]2^{1/2} [/pmath]

(B) [pmath]3^{1/3} [/pmath]

(C) [pmath]4^{1/4} [/pmath]

(D) [pmath]5^{1/5} [/pmath]

(E) [pmath]6^{1/6}[/pmath]

Answer

One way to break down this problem is to compare the quantities a pair at a time. First, consider [pmath]2^{1/2}[/pmath] and [pmath]3^{1/3}[/pmath]. Write them next to each other, with a question mark between.

[pmath]2^{1/2}[/pmath] ? [pmath]3^{1/3}[/pmath]

The question mark stands for > or < , since one of them is almost certainly going to be larger. To compare these quantities exactly, we can raise them both to the same power, since their relative order will be preservedthat is, whichever one was larger to begin with will still be larger. (This works if both numbers are positive.)

Raising both numbers to the [pmath]6^th[/pmath] power gets rid of both fractions in the exponents.

[pmath](2^{1/2})^6[/pmath] ? [pmath](3^{1/3})^6[/pmath]

[pmath]2^3[/pmath] ? [pmath]3^2[/pmath]

Now we can easily evaluate: [pmath]2^3[/pmath] = 8, whereas [pmath]3^2[/pmath] = 9. This tells us that

[pmath]2^{1/2}[/pmath] < [pmath]3^{1/3}[/pmath].

(B) is larger than (A). Rule out (A).

Answer choice (C), [pmath]4^{1/4}[/pmath], is the same as [pmath]{2^2}^{1/4}[/pmath] = [pmath]2^{1/2}[/pmath], which is the same as (A). So we can rule out (C) along with (A).

We can continue with pairwise comparisons by raising both expressions to a larger power that eliminates the fractional exponent.Compare (B) and (D) by raising both to the [pmath]15^th[/pmath] power.

[pmath](3^{1/3})^15[/pmath] ? [pmath](5^{1/5})^15[/pmath]

[pmath]3^5[/pmath] ? [pmath]5^3[/pmath]

243 > 125

(B) is still larger.

Finally, compare (B) and (E) by raising both to the [pmath]6^th[/pmath] power.

[pmath](3^{1/3})^6[/pmath] ? [pmath](6^{1/6})^6[/pmath]

[pmath]3^2[/pmath] ? [pmath]6^1[/pmath]

9 > 6

(B) is still larger. Since it is larger than every other choice, it is the answer.

Alternatively, we could have raised every choice to the same large power (say, the [pmath]60^th[/pmath]) to eliminate all the fractional exponents in one fell swoop. However, we would wind up with [pmath]2^30[/pmath], [pmath]3^20[/pmath], [pmath]4^15[/pmath], [pmath]5^12[/pmath], and [pmath]6^10[/pmath], and we would still need to do some pairwise comparisons.

The correct answer is B.

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