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Prove that $\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{ab}$
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Proof

If we multiply the expression on the left hand side by itself $n$ times, we have

$(\sqrt[n]{a} \sqrt[n]{b})^n = (\sqrt[n]{a})^n (\sqrt[n]{b})^n = ab$.

On the other hand, a positive number ab has only a single positive root, $\sqrt[n]{ab}$.

Therefore,

$\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{ ab}$.

This completes our proof.

by Diamond (55,292 points)