Answer
The number to the nearest three significant digits is $37.0\times {{10}^{-9}}$.
Work Step by Step
$\frac{\left( 612\times {{10}^{-6}} \right)\left( 15\times {{10}^{-9}} \right)\left( 2.7\times {{10}^{3}} \right)}{\left( 82\times {{10}^{9}} \right)\left( 8.16\times {{10}^{-12}} \right)}$,
Combine the powers of $10$ by using the formula $\frac{{{a}^{x}}\cdot {{a}^{y}}}{{{a}^{z}}}={{a}^{x+y-z}}$.
$\begin{align}
& \frac{\left( 612\times {{10}^{-6}} \right)\left( 15\times {{10}^{-9}} \right)\left( 2.7\times {{10}^{3}} \right)}{\left( 82\times {{10}^{9}} \right)\left( 8.16\times {{10}^{-12}} \right)}=\frac{612\times 15\times 2.7\times {{10}^{-6-9+3}}}{82\times 8.16\times {{10}^{9-12}}} \\
& =\frac{612\times 15\times 2.7\times {{10}^{-12}}}{82\times 8.16\times {{10}^{-3}}} \\
& =\frac{612\times 15\times 2.7\times {{10}^{-12+3}}}{82\times 8.16}
\end{align}$
Further simplify the above expression.
$\begin{align}
& \frac{\left( 612\times {{10}^{-6}} \right)\left( 15\times {{10}^{-9}} \right)\left( 2.7\times {{10}^{3}} \right)}{\left( 82\times {{10}^{9}} \right)\left( 8.16\times {{10}^{-12}} \right)}=\frac{612\times 15\times 2.7\times {{10}^{-12+3}}}{82\times 8.16} \\
& =37.04\times {{10}^{-9}}
\end{align}$
