Determine the ratio of the volumes of the two similar spheres.

Let $V_S$ be the volume of the smaller sphere and $V_L$ be the volume of the larger sphere.
$V_S = 1408 \text{ cm}^3$
$V_L = 4752 \text{ cm}^3$
The ratio of the volumes (smaller to larger) is:

$$\frac{V_S}{V_L} = \frac{1408}{4752}$$

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 176:

$$\frac{1408 \div 176}{4752 \div 176} = \frac{8}{27}$$

So, the volume scale factor (VSF) from the smaller to the larger sphere is $\frac{8}{27}$.

The linear scale factor is the cube root of the volume scale factor:

$$LSF = \sqrt[3]{\frac{V_S}{V_L}} = \sqrt[3]{\frac{8}{27}}$$ $$LSF = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}$$

This means that the linear dimensions (like radii) of the smaller sphere are $\frac{2}{3}$ times those of the larger sphere.

The area scale factor is the square of the linear scale factor:

$$ASF = (LSF)^2 = \left(\frac{2}{3}\right)^2$$ $$ASF = \frac{2^2}{3^2} = \frac{4}{9}$$

This means that the surface area of the smaller sphere is $\frac{4}{9}$ times the surface area of the larger sphere.

Let $A_S$ be the surface area of the smaller sphere and $A_L$ be the surface area of the larger sphere.
We are given $A_S = 352 \text{ cm}^2$.
Using the area scale factor:

$$\frac{A_S}{A_L} = ASF$$ $$\frac{352}{A_L} = \frac{4}{9}$$

To find $A_L$, rearrange the equation:

$$A_L = 352 \times \frac{9}{4}$$ $$A_L = \frac{352 \times 9}{4}$$ $$A_L = 88 \times 9$$ $$A_L = 792 \text{ cm}^2$$

The surface area of the larger sphere is:

$$\text{792 cm}^2$$