swerve wrote:Which of the following quantities is the largest?
A. \(\sqrt{2}\)
B. \(\sqrt[3]{3}\)
C. \(\sqrt[4]{4}\)
D. \(\sqrt[5]{5}\)
E. \(\sqrt[6]{6}\)
The OA is B
Source: Manhattan Prep
Note that we have to deal with exponents wich are in fraction. If base increases, its exponent decreases; thus, we cannot outrightly reject any option.
So, we have five numbers: 2^(1/2); 3^(1/3); 4^(1/4); 5^(1/5); and 6^(1/6)
The easiest two to compares are 2^(1/2) and 4^(1/4). Let's write 4^(1/4) = [(2)^2]^(1/4) = 2^(1/2). So 2^(1/2) and 4^(1/4) are equal; thus, Option A and C are ruled out.
So, we have three numbers 3^(1/3); 5^(1/5); and 6^(1/6) to choose from.
To compare the above numbers, either bases or the exponents should be equal. The easiest is to equate exponents. Let's make LCM of the denominators of exponents 1/3, 1/5 and 1/6 as the common exponents. LCM = 30
Thus,
3^(1/3) = 3^(10/30);
5^(1/5) = 5^(6/30);
6^(1/6) = 6^(5/30)
Since 1/30 is common to all, we can ignore it. So, the numbers are 3^10; 5^6 and 6^5. Let's compare 3^10 and 6^5. Since 3^10 = (3^2)^5 = 9^5, we can reject 6^5 as 9^5 > 6^5.
So, the last two are 3^10 and 5^6. We have 3^10 = (3^5)^2 = 243^2 and 5^6 = (5^3)^2 = 125^2. It is clear that 243^2 > 125^2. Thus, 3^(1/3) is the largest.
The correct answer:
B
Hope this helps!
-Jay
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