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Which of the following quantities is the largest?

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Which of the following quantities is the largest?  [#permalink]

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New post Updated on: 27 Dec 2013, 04:27
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Question Stats:

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Which of the following quantities is the largest?

(A) \(\sqrt{2}\)

(B) \(\sqrt[3]{3}\)

(C) \(\sqrt[4]{4}\)

(D) \(\sqrt[5]{5}\)

(E) \(\sqrt[6]{6}\)

Originally posted by mrinal2100 on 14 Jun 2011, 09:22.
Last edited by Bunuel on 27 Dec 2013, 04:27, edited 2 times in total.
Renamed the topic, edited the question added the answer choices and OA.
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Re: number  [#permalink]

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New post 14 Jun 2011, 19:08
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mrinal2100 wrote:
sorry guys for the wron quote but the choices were
a)2(pow)(1/2)
b)3(pow)(1/3)
c)4(pow)(1/4)
d)5(pow)(1/5)
e)6(pow)(1/6)

its not 6*(sqroot of 6)


So I am assuming that the choices are:
\(2^{\frac{1}{2}}, 3^{\frac{1}{3}}, 4^{\frac{1}{4}}, 5^{\frac{1}{5}}, 6^{\frac{1}{6}}\)

Since fractional powers are a pain, let me multiply all the powers by 60 (LCM) to make them manageable.

\(2^{30}, 3^{20}, 4^{15} ( = 2^{30}), 5^{12}, 6^{10}\)
Bases and powers, both are different. To compare, we need to make one of them the same.

\(2^{30} = 8^{10}\)
\(3^{20} = 9^{10}\)
\(6^{10}\)

Obviously, out of these three, \(9^{10}\) is greatest.

Now we just need to compare \(3^{20}\) with \(5^{12}\)
\(3^{20} = 243^{4}\)
\(5^{12} = 125^{4}\)

\(3^{20} ( = 3^{\frac{1}{3}})\) is the greatest.
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Re: Which of the following quantities is the largest?  [#permalink]

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New post 27 Dec 2013, 04:20
mrinal2100 wrote:
Which of the following quantities is the largest?

(A) \(\sqrt{2}\)

(B) \(\sqrt[3]{3}\)

(C) \(\sqrt[4]{5}\)

(D) \(\sqrt[5]{5}\)

(E) \(\sqrt[6]{6}\)


The question shows another answer choice for C, but I second your approach Karishma

Now, let's assume the question as is

2^30, 3^20 and 5^15

In this case, I assume it would be a matter of setting all exponents to GCF and compare the bases

So we would end up with 64^5, 81^5, 64^5

Clearly now B is the winner here

Kudos rain!

Cheers!
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Re: Which of the following quantities is the largest?  [#permalink]

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New post 27 Dec 2013, 04:48
jlgdr wrote:
mrinal2100 wrote:
Which of the following quantities is the largest?

(A) \(\sqrt{2}\)

(B) \(\sqrt[3]{3}\)

(C) \(\sqrt[4]{5}\)

(D) \(\sqrt[5]{5}\)

(E) \(\sqrt[6]{6}\)


The question shows another answer choice for C, but I second your approach Karishma

Now, let's assume the question as is

2^30, 3^20 and 5^15

In this case, I assume it would be a matter of setting all exponents to GCF and compare the bases

So we would end up with 64^5, 81^5, 64^5

Clearly now B is the winner here

Kudos rain!

Cheers!
J :)


Edited option C. It should be as written in Karishma's post.
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Re: Which of the following quantities is the largest?  [#permalink]

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New post 30 Dec 2013, 23:26
1
jlgdr wrote:
Now, let's assume the question as is

2^30, 3^20 and 5^15

In this case, I assume it would be a matter of setting all exponents to GCF and compare the bases

So we would end up with 64^5, 81^5, 64^5

Clearly now B is the winner here

Kudos rain!

Cheers!
J :)


2^30, 3^20 and 5^15
Yes, you can get the common power of 5

64^5, 81^5 and 125^5

So 5^15 is greatest here.
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Re: Which of the following quantities is the largest?  [#permalink]

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New post 22 Oct 2015, 11:56
VeritasPrepKarishma wrote:
mrinal2100 wrote:
sorry guys for the wron quote but the choices were
a)2(pow)(1/2)
b)3(pow)(1/3)
c)4(pow)(1/4)
d)5(pow)(1/5)
e)6(pow)(1/6)

its not 6*(sqroot of 6)


So I am assuming that the choices are:
\(2^{\frac{1}{2}}, 3^{\frac{1}{3}}, 4^{\frac{1}{4}}, 5^{\frac{1}{5}}, 6^{\frac{1}{6}}\)

Since fractional powers are a pain, let me multiply all the powers by 60 (LCM) to make them manageable.

\(2^{30}, 3^{20}, 4^{15} ( = 2^{30}), 5^{12}, 6^{10}\)
Bases and powers, both are different. To compare, we need to make one of them the same.

\(2^{30} = 8^{10}\)
\(3^{20} = 9^{10}\)
\(6^{10}\)

Obviously, out of these three, \(9^{10}\) is greatest.

Now we just need to compare \(3^{20}\) with \(5^{12}\)
\(3^{20} = 243^{4}\)
\(5^{12} = 125^{4}\)

\(3^{20} ( = 3^{\frac{1}{3}})\) is the greatest.



I Karishma - it's very clear and simple until the end. To compare 3^20 with 5^12, it seems that we need to know what 3^16 and 5^8 are. Isn't 3^16 a uncommon number to memorize? Or is this expected for the gmat / there is an easier way to compare? Thanks so much!
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Re: Which of the following quantities is the largest?  [#permalink]

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New post 22 Oct 2015, 12:08
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happyface101 wrote:
VeritasPrepKarishma wrote:
mrinal2100 wrote:
sorry guys for the wron quote but the choices were
a)2(pow)(1/2)
b)3(pow)(1/3)
c)4(pow)(1/4)
d)5(pow)(1/5)
e)6(pow)(1/6)

its not 6*(sqroot of 6)


So I am assuming that the choices are:
\(2^{\frac{1}{2}}, 3^{\frac{1}{3}}, 4^{\frac{1}{4}}, 5^{\frac{1}{5}}, 6^{\frac{1}{6}}\)

Since fractional powers are a pain, let me multiply all the powers by 60 (LCM) to make them manageable.

\(2^{30}, 3^{20}, 4^{15} ( = 2^{30}), 5^{12}, 6^{10}\)
Bases and powers, both are different. To compare, we need to make one of them the same.

\(2^{30} = 8^{10}\)
\(3^{20} = 9^{10}\)
\(6^{10}\)

Obviously, out of these three, \(9^{10}\) is greatest.

Now we just need to compare \(3^{20}\) with \(5^{12}\)
\(3^{20} = 243^{4}\)
\(5^{12} = 125^{4}\)

\(3^{20} ( = 3^{\frac{1}{3}})\) is the greatest.



I Karishma - it's very clear and simple until the end. To compare 3^20 with 5^12, it seems that we need to know what 3^16 and 5^8 are. Isn't 3^16 a uncommon number to memorize? Or is this expected for the gmat / there is an easier way to compare? Thanks so much!


It is not \(3^{16}\) that you need to remember.

When you end up comparing \(3^{20}\) and \(5^{12}\), try to raise the 2 numbers to the same power.

In this case the common GCD of 20 and 12 is 4.

Thus \(3^{20}=(3^5)^4\) and

\(5^{12} = (5^3)^4\)

Giving you the 2 values as \(243^4\) and \(125^4\). So now can easily see that the 2 numbers are raised to the same power but with a differnet base giving you \(3^{20} > 5^{12}\)
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Re: Which of the following quantities is the largest?  [#permalink]

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New post 19 Feb 2018, 14:33
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Re: Which of the following quantities is the largest? &nbs [#permalink] 19 Feb 2018, 14:33
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