The volume of a cubical box is 474.552 cubic metres.

Solution:

Volume of a cube is given by:

$V=s^{3}$, where $s=$ side of the cube

Now

$s^{3}=474.552$ cubic metres

$\Rightarrow s=\sqrt[3]{474.552}=\sqrt[3]{\frac{474552}{1000}}=\frac{\sqrt[3]{474552}}{\sqrt[3]{1000}}$

To find the cube root of 474552, we need to proceed as follows:

On factorising 474552 into prime factors, we get:

$474552=2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 13 \times 13 \times 13$

On grouping the factors in triples of equal factors, we get:

$474552=\{2 \times 2 \times 2\} \times\{3 \times 3 \times 3\} \times\{13 \times 13 \times 13\}$

Now, taking one factor from each triple, we get:

$\sqrt[3]{474552}=\sqrt[3]{\{2 \times 2 \times 2\} \times\{3 \times 3 \times 3\} \times\{13 \times 13 \times 13\}}=2 \times 3 \times 13=78$

Also,

$\sqrt[3]{1000}=10$

$\therefore s=\frac{\sqrt[3]{474552}}{\sqrt[3]{1000}}=\frac{78}{10}=7.8$

Thus, the length of the side is 7.8 m.