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Which of the following quantities is the largest?

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by Jay@ManhattanReview » Sun Aug 04, 2019 11:33 pm
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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by Scott@TargetTestPrep » Sun Aug 11, 2019 6:11 pm
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
First, we notice that fourth root of 4 is the same thing as the square root of 2. If that's hard to see, we can write: 4^(1/4) = (2^2)^(1/4) = 2^(2/4) = 2^(1/2). Therefore, neither A nor C can be the correct answer and we eliminate them both.

Next, notice that 3^(1/3) = (3^2)^(1/6) = 9^(1/6). Since 9 > 6, 9^(1/6) is greater than 6^(1/6). We eliminate E as well.

To decide between B and D, let's compare the 15th power of 3^(1/3) and 5^(1/5).

[3^(1/3)]^15 = 3^5 = 243

[5^(1/5)]^15 = 5^3 = 125

We see that 3^(1/3) is the greatest among the given expressions.

Answer: B

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