Answer: 6 zeros

Multiplying by $10$, or multiples/powers of $10$, makes zeros, so find powers of $10$

    $3^4\times 4^5\times 5^6$

$=3^4\times 2^{10}\times 5^6$

$=3^4\times 2^4\times 2^6 \times 5^6$

$=\underbrace{6^4}_{\text{no zeros}}\times\underbrace{10^6}_{\text{six zeros}}$