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

Question

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.

Bunuel · Answer

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}
eq{integer} , so in this case the answer is NO. Not sufficient.

Answer: A.

cyberjadugar · Answer

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.

Thus, Answer is (A)

Regards,

GMATinsight · Answer

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.

1) √m = Ineteger
i.e.  (\sqrt{m})^n = Integer^{Integer}  
But  Integer^{Integer} = Integer 
SUFFICIENT

2)√n = integer does not confirm whether n is even (Yes) or not (No) hence
NOT SUFFICIENT

Answer: Option A

Basshead · Answer

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

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

Integer^{integer}  = integer. SUFFICIENT.

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

We can get a yes and no answer here because we don't know if  \sqrt{m}  is an integer. INSUFFICIENT.

Answer is A.

nik256 · Answer

here we know both m and n are positive integers

we need to find if this is an integer 
 (\sqrt{m})^n

Lets look at statement 1

root m is an integer

so we know (int)^(int) is an integer

So this is sufficient as (root m)^n is an integer according to this

Looking at st 2 we know that root n is integer but this statement doesnt help as we know nothing about root m'

so this st is insufficient

Answer (A)

bumpbot · Answer

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

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