A square based brass plate is 2mm high and has a mass of 1.05kg. The density of the brass is 8.4g/cm³. Calculate the length of the plate in centimeters.

The density is given in g/cm³, so convert mass to grams and height to centimeters.
Given mass ($m$) = $1.05 \text{ kg}$.

$$m = 1.05 \text{ kg} \times \frac{1000 \text{ g}}{1 \text{ kg}} = 1050 \text{ g}$$

Given height ($h$) = $2 \text{ mm}$.

$$h = 2 \text{ mm} \times \frac{1 \text{ cm}}{10 \text{ mm}} = 0.2 \text{ cm}$$

Given density ($\rho$) = $8.4 \text{ g/cm}^3$.

$$V = \frac{m}{\rho}$$

Substitute the converted mass and given density:

$$V = \frac{1050 \text{ g}}{8.4 \text{ g/cm}^3}$$ $$V = 125 \text{ cm}^3$$

For a square base with side length $L$, the area of the base is $L^2$. The volume is the base area multiplied by the height.

$$V = L^2 \times h$$

Rearrange the formula to solve for $L^2$:

$$L^2 = \frac{V}{h}$$

Substitute the calculated volume and the converted height:

$$L^2 = \frac{125 \text{ cm}^3}{0.2 \text{ cm}}$$ $$L^2 = 625 \text{ cm}^2$$

$$L = \sqrt{625 \text{ cm}^2}$$ $$L = 25 \text{ cm}$$

The length of the plate is $25 \text{ cm}$.