Use indices (including negative and fractional indices) and the index laws.

Simplify $\left(\frac{8x^{-2}}{y^{3/2}}\right)^{1/3}$.

A population of bacteria grows according to the model \(P(t) = P_0 \cdot 2^{kt}\), where \(P(t)\) is the population at time \(t\) hours, \(P_0\) is the initial population, and \(k\) is a constant. The table below shows the population at selected times. (a) Use the information in the table to determine the value of \(k\). [2 marks] (b) Use your answer from part (a) to predict the population at \(t = 9\) hours. Express your answer in the form \(a \times 2^b\), where \(a\) and \(b\) are integers. [2 marks]

A biotechnology company models the growth rate of two bacterial colonies under different nutrient conditions. The table in the stimulus shows the population P (measured in thousands of cells) of Colony X at various time points, where t is measured in hours. The population of Colony Y at time t hours is modelled by the function Q(t) = 8 × 3^(t/4). (a) Use the data in the table to determine the value of k if the population of Colony X is modelled by P(t) = k × 2^(t/3). Give your answer correct to two decimal places. [2 marks] (b) Use your answer from part (a) to determine the population of Colony X at t = 10 hours. Give your answer in thousands of cells, correct to one decimal place. [1 mark] (c) Determine the time t at which both colonies have the same population. Give your answer correct to two decimal places. [2 marks]