28 (a) Simplify: (i) $h^3 \times h^3$ (ii) $\frac{f^9}{f^3}$ (b) The length of each side of a plastic cube is 2a millimetres - OCR - GCSE Maths - Question 28 - 2019 - Paper 1

To simplify this expression, we subtract the exponents in accordance with the property of exponents for division:

$\frac{f^9}{f^3} = f^{9-3} = f^6$

The density ($\rho$) of an object is given by the formula:

$\rho = \frac{\text{mass}}{\text{volume}}$

For the cube, the volume can be calculated as:

$\text{Volume} = (\text{side length})^3 = (2a)^3 = 8a^3$

Thus, substituting in the values:

$\rho = \frac{32a^2 \text{ grams}}{8a^3}$

This simplifies to:

$\rho = \frac{32}{8} \cdot \frac{a^2}{a^3} = 4 \cdot \frac{1}{a} = \frac{4}{a} \text{ grams per millimetre cubed (g/mm}^3\text{)}$

The units for density based on the mass in grams and the volume in millimetres cubed are:

grams per millimetre cubed (g/mm³).