1. 1
   A textile designer scales up a square patch of fabric with side length $5x^3$.

   (a)

   Simplify the expression for the area, $(5x^3)^2$.

   [1 mark]

   • A$25x^6$
   • B$25x^5$
   • C$10x^6$
   • D$5x^6$

   (b)

   The designer also needs to simplify a ratio of two fabric lengths, $12a^5 \div 3a^{-2}$. What is the simplified result?

   [1 mark]

   • A$4a^3$
   • B$4a^7$
   • C$9a^7$
   • D$4a^{-10}$

   (c)

   Finally, the designer combines two panel measurements given by $6x^7y^4 \times 5x^{-5}y$. Which expression is the fully simplified product?

   [1 mark]

   • A$30x^{12}y^4$
   • B$11x^2y^5$
   • C$30x^2y^5$
   • D$30x^2y^4$

   Total for question 1: 3 marks

2. 2
   A computing student is comparing storage capacities expressed algebraically.

   (a)

   Simplify $(2p^4)^3$, giving your answer in the form $kp^n$.

   [2 marks]

   (b)

   The student then simplifies $\frac{20q^9}{4q^{-3}}$, giving the answer in the form $kq^n$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An engineer scales a component and needs to check a design formula involving powers.

   (a)

   An engineer scales a component and needs to simplify $\frac{(3m^2)^3 \times 2m}{6m^4}$. Show your working and give the answer in the form $km^n$.

   [3 marks]

   (b)

   The engineer's design formula requires solving $2^{x} = 32$ for $x$. Solve this equation, showing your method.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A scientist models bacterial culture volume using indices to compare growth rates of two cultures.

   (a)

   A scientist models bacterial culture volume using indices. Simplify the expression $\frac{(4n^3)^2}{2n^{-1}} \times n^2$, giving your answer in the form $kn^p$.

   [4 marks]

   (b)

   The scientist compares this culture to a second culture whose volume simplifies to $8n^9 \div 4n^3$. Simplify this expression fully.

   [4 marks]

   (c)

   The scientist's growth model leads to the equation $5^{x+1} = 25^x$. Solve this equation for $x$, showing full working.

   [5 marks]

   Total for question 4: 13 marks

End of questions