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

To simplify the fraction, we use the property that states $\frac{a^m}{a^n} = a^{m-n}$. Therefore:

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

Density is defined as mass divided by volume. We have:

1. The mass of the cube is $32a^2$ grams.
2. The volume of a cube is given by the formula $V = \text{side length}^3$. Since the side length is $2a$ millimetres: $V = (2a)^3 = 8a^3$\

Now, we can express density ($\rho$) as:

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

Simplifying that expression yields:

$\rho = \frac{32}{8} \cdot \frac{a^2}{a^3} = 4 \cdot \frac{1}{a} = \frac{4}{a}$

The mass is in grams and the volume is in cubic millimetres. Therefore, the units for density will be:

$\text{grams per cubic millimetre (g/mm}^3)$