How do you negate the following sentence?

Question

$ ∀x ∈ R,∃n ∈ Z ,x^{n}>0 $

How do you negate it so that the ¬ symbol does not arise to the left of any quantifier? Is the negated statement is true?

Answer 1

How do you negate it so that the ¬ symbol does not arise to the left of any quantifier?

Let $P$ be some property on a variable $z .$

Two important hints:

$$ \tag{1}\neg [\forall z, P(z)] \iff \exists z: \neg P(z)$$

$$ \tag{2} \neg [\exists z : P(z)] \iff \forall z, \neg P(z)$$

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Is the negated statement true?

Since telling you the negation would be spoiling things for you, note that a statement is true if and only if its negation is false (i.e. precisely one of either a statement or its negation must be true).

The statement you've given is: "Given any real number $x$, there's some integer $n$ such that $x^n$ is positive."

You tell me: for $x=0,$ does there exist such an $n \in \mathbb{Z}$?