Three cuboids of dimensions 5 cm × 6 cm × 7cm,

The dimensions of the three cuboids are $5 \mathrm{~cm} \times 6 \mathrm{~cm} \times 7 \mathrm{~cm}, 4 \mathrm{~cm} \times 7 \mathrm{~cm} \times 8 \mathrm{~cm}$ and $2 \mathrm{~cm} \times 3 \mathrm{~cm} \times 13 \mathrm{~cm}$.

Now, a new cube is formed by melting the given cuboids.

$\therefore$ Voulume of the cube $=$ sum of the volumes of the cuboids

$=(5 \mathrm{~cm} \times 6 \mathrm{~cm} \times 7 \mathrm{~cm})+(4 \mathrm{~cm} \times 7 \mathrm{~cm} \times 8 \mathrm{~cm})+(2 \mathrm{~cm} \times 3 \mathrm{~cm} \times 13 \mathrm{~cm})$

$=\left(210 \mathrm{~cm}^{3}\right)+\left(224 \mathrm{~cm}^{3}\right)+\left(78 \mathrm{~cm}^{3}\right)$

$=512 \mathrm{~cm}^{3}$

Since volume of a cube $=(\text { side })^{3}$, we have:

$512=(\text { side })^{3}$

$\Rightarrow($ side $)=\sqrt[3]{512}=8 \mathrm{~cm}$

$\therefore$ The side of the new cube is $8 \mathrm{~cm}$.