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

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

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

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

Diagnostic Test
Question: 44
Page: 26
Difficulty: 600

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

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