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

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Question Stats: 39% (01:59) correct 61% (01:20) wrong based on 417 sessions

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Which of the following quantities is the largest?

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

(B) $$\sqrt{3}$$

(C) $$\sqrt{4}$$

(D) $$\sqrt{5}$$

(E) $$\sqrt{6}$$

Originally posted by mrinal2100 on 14 Jun 2011, 10:22.
Last edited by Bunuel on 27 Dec 2013, 05:27, edited 2 times in total.
Renamed the topic, edited the question added the answer choices and OA.
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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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mrinal2100 wrote:
Which of the following quantities is the largest?

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

(B) $$\sqrt{3}$$

(C) $$\sqrt{5}$$

(D) $$\sqrt{5}$$

(E) $$\sqrt{6}$$

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 Math Expert V
Joined: 02 Sep 2009
Posts: 58443
Re: Which of the following quantities is the largest?  [#permalink]

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jlgdr wrote:
mrinal2100 wrote:
Which of the following quantities is the largest?

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

(B) $$\sqrt{3}$$

(C) $$\sqrt{5}$$

(D) $$\sqrt{5}$$

(E) $$\sqrt{6}$$

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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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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Intern  Joined: 05 Aug 2015
Posts: 40
Re: Which of the following quantities is the largest?  [#permalink]

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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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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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_________________ Re: Which of the following quantities is the largest?   [#permalink] 20 Apr 2019, 23:59
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