= Solution
If $p$ divides $a$ or $b$, it divides their product and both sides of the proposed identity are zero. Otherwise use <Euler's criterion> and the ordinary exponent rule:
$$
\left(\frac{ab}p\right)\equiv(ab)^{(p-1)/2}
\equiv a^{(p-1)/2}b^{(p-1)/2}
\equiv\left(\frac ap\right)\left(\frac bp\right)\pmod p.
$$
Both sides are in $\{1,-1\}$, and these two values are distinct modulo the odd prime $p$. Hence the congruence is an equality:
$$
\boxed{\left(\frac{ab}p\right)=\left(\frac ap\right)\left(\frac bp\right).}
$$
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