1. 1
   $a=2^3\times2^4$; $b=2^{10}\div2^6$; $c=(2^2)^5$.

   (a)

   Write $a$ as a single power of 2.

   [1 mark]

   • A$2^1$
   • B$2^{12}$
   • C$2^7$
   • D$2^{-1}$

   (b)

   Find the value of $b$.

   [1 mark]

   • A4
   • B16
   • C64
   • D256

   (c)

   Find the value of $c$.

   [1 mark]

   • A32
   • B100
   • C20
   • D1024

   Total for question 1: 3 marks

2. 2
   A country's population is $4.5\times10^7$. A city within the country has a population of $9\times10^5$.

   (a)

   Find how many times bigger the country's population is than the city's population, giving your answer as an ordinary number.

   [2 marks]

   (b)

   Write the combined population of the country and city in standard form.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A model-making company scales its measurements using fractional and negative indices. Two of the scaling factors used are $16^{\frac{1}{4}}$ and $27^{\frac{2}{3}}$.

   (a)

   Evaluate $16^{\frac{1}{4}}$ and $27^{\frac{2}{3}}$, showing your method.

   [3 marks]

   (b)

   Evaluate $8^{-\frac{2}{3}}$, and hence find the value of $8^{-\frac{2}{3}} \times 27^{\frac{2}{3}}$ using your answer to part (a).

   [3 marks]

   Total for question 3: 6 marks

4. 4
   Light travels at a speed of $3\times10^8$ m/s. The distance from the Sun to a distant planet is $4.5\times10^{12}$ m. A spacecraft travels at a constant speed of $2.5\times10^4$ m/s.

   (a)

   Calculate how long light takes to travel from the Sun to the planet, giving your answer in seconds in standard form.

   [4 marks]

   (b)

   Calculate how long the spacecraft would take to travel the same distance, giving your answer in seconds in standard form.

   [4 marks]

   (c)

   Find how many times longer the spacecraft's journey time is than light's journey time, and convert the spacecraft's journey time into years, giving your answer correct to 3 significant figures (take 1 year $= 3.15\times10^7$ seconds).

   [5 marks]

   Total for question 4: 13 marks

End of questions