Express $\forall n \in \mathbb N$, $\exists m \in \mathbb N$, $n^4 = m^2$ in words without using the symbol $\mathbb N$

Question

Express $\forall n \in \mathbb N$, $\exists m \in \mathbb N$, $n^4 = m^2$ in words without using the symbol $\mathbb N$.

My Solution:

For all $n$ that is an element of Natural number there is $m$ that is also an element of natural number such that $n^4 = m^2.$

Can I have a feed back on my solution or explanation on how to do the problem.

Answer 1

Your translation is equivalent to the symbolic statement, and you omitted the symbol $\mathbb N$. So it successfully satisfies (answers) the question.

But you might want to take the next step to try "naturalize" the statement into natural language:

"For every natural number $n$, there is a natural number $m$ such that $n^4 = m^2$."

Or, more simply yet: "Every perfect fourth power is also a perfect square."

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Can you see that the statement is true?

For every $n\in \mathbb N$, choose $m = n^2$. Choose any $n \in \mathbb N$. For this choice, put $m = n^2$. Then $n^4 = (n^2)^2 = m^2$.