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# If m and n are positive integers, is root(m)^n an integer?

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Math Expert
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If m and n are positive integers, is root(m)^n an integer? [#permalink]

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09 Jul 2012, 04:56
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Difficulty:

5% (low)

Question Stats:

71% (01:35) correct 29% (00:47) wrong based on 479 sessions

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If m and n are positive integers, is $$(\sqrt{m})^n$$ an integer?

(1) $$\sqrt{m}$$ is an integer.
(2) $$\sqrt{n}$$ is an integer.

Diagnostic Test
Question: 44
Page: 26
Difficulty: 600
[Reveal] Spoiler: OA

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Math Expert
Joined: 02 Sep 2009
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Kudos [?]: 68366 [0], given: 9797

Re: If m and n are positive integers, is root(m)^n an integer? [#permalink]

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09 Jul 2012, 04:56
Expert's post
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SOLUTION

If m and n are positive integers, is $$(\sqrt{m})^n$$ an integer?

(1) $$\sqrt{m}$$ is an integer --> $$(\sqrt{m})^n=(integer)^{positive \ integer}=integer$$. Sufficient.

(2) $$\sqrt{n}$$ is an integer. If $$m=n=1$$, then the answer is YES but if $$m=2$$ and $$n=1$$, then $$(\sqrt{m})^n=\sqrt{2}\neq{integer}$$, so in this case the answer is NO. Not sufficient.

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Re: If m and n are positive integers, is root(m)^n an integer? [#permalink]

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09 Jul 2012, 05:27
Bunuel wrote:
If m and n are positive integers, is $$(\sqrt{m})^n$$ an integer?

(1) $$\sqrt{m}$$ is an integer.
(2) $$\sqrt{n}$$ is an integer.

Hi,

Difficulty level: 600

We have to find whether $$(\sqrt{m})^n$$ is an integer,
or $$m^{\frac n2}$$ is an integer.

Given, m & n are integers,

Using (1),
$$\sqrt{m}$$ is an integer
then, $$(integer)^{(integer)}$$ is Integer. Sufficient.

Using (2),
$$\sqrt{n}$$ is an integer, or n is an integer.
lets say, m=3, n=9,
$$(\sqrt{m})^n$$=$$(\sqrt{3})^9$$ which is not an integer. Insufficient.

Regards,
Math Expert
Joined: 02 Sep 2009
Posts: 32549
Followers: 5636

Kudos [?]: 68366 [0], given: 9797

Re: If m and n are positive integers, is root(m)^n an integer? [#permalink]

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13 Jul 2012, 04:09
Expert's post
SOLUTION

If m and n are positive integers, is $$(\sqrt{m})^n$$ an integer?

(1) $$\sqrt{m}$$ is an integer --> $$(\sqrt{m})^n=(integer)^{positive \ integer}=integer$$. Sufficient.

(2) $$\sqrt{n}$$ is an integer. If $$m=n=1$$, then the answer is YES but if $$m=2$$ and $$n=1$$, then $$(\sqrt{m})^n=\sqrt{2}\neq{integer}$$, so in this case the answer is NO. Not sufficient.

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Re: If m and n are positive integers, is root(m)^n an integer? [#permalink]

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17 Jul 2014, 20:49
Hello from the GMAT Club BumpBot!

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Re: If m and n are positive integers, is root(m)^n an integer? [#permalink]

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21 Dec 2015, 10:45
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If m and n are positive integers, is root(m)^n an integer?   [#permalink] 21 Dec 2015, 10:45
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