1. **State the problem:** We have two mathematically similar cuboids. The smaller cuboid has a shortest edge of 5 cm and a surface area of 400 cm^2. The larger cuboid has a shortest edge of 8 cm. We need to find the surface area of the larger cuboid. 2. **Recall the formula and rules:** For similar 3D shapes, the ratio of corresponding linear dimensions (edges) is the scale factor $k$. The ratio of surface areas is the square of the scale factor: $$\frac{\text{Surface Area}_2}{\text{Surface Area}_1} = k^2$$ 3. **Calculate the scale factor:** $$k = \frac{\text{shortest edge of larger cuboid}}{\text{shortest edge of smaller cuboid}} = \frac{8}{5}$$ 4. **Calculate the surface area of the larger cuboid:** Using the surface area ratio, $$\frac{\text{Surface Area}_2}{400} = \left(\frac{8}{5}\right)^2$$ $$\text{Surface Area}_2 = 400 \times \left(\frac{8}{5}\right)^2$$ 5. **Simplify:** $$\text{Surface Area}_2 = 400 \times \frac{64}{25}$$ $$\text{Surface Area}_2 = 400 \times 2.56 = 1024$$ **Final answer:** The surface area of the larger cuboid is $1024$ cm$^2$.