Cube Number multiplied by Cube Number is Cube

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Let $a, b \in \N$ be natural numbers.

Let $a$ and $b$ be cube numbers.

Then $a b$ is also a cube number.

In the words of Euclid:

If a cube number by multiplying a cube number make some number the product will be cube.

(The Elements: Book $\text{IX}$: Proposition $4$)


By the definition of cube number:

$\exists r \in \N: r^3 = a$
$\exists s \in \N: s^3 = b$


\(\ds a b\) \(=\) \(\ds \paren {r^3} \paren {s^3}\) Power of Product
\(\ds \) \(=\) \(\ds \paren {r s}^3\)


$\exists k = r s \in \N: a = k^3$

Hence the result by definition of cube number.


Historical Note

This proof is Proposition $4$ of Book $\text{IX}$ of Euclid's The Elements.
The proof as given here is not that given by Euclid, as the latter is unwieldy and of limited use.