Laws of exponentsIB MYP Maths Extended: Subtopic test

1. 1
   Let $x$ be any non-zero number.

   (a)

   Simplify $x^7\times x^{-3}$.

   [1 mark]

   • A$x^{10}$
   • B$x^{-21}$
   • C$x^{4}$
   • D$x^{21}$

   (b)

   Simplify $(x^3)^{-2}$.

   [1 mark]

   • A$\frac{1}{x^6}$
   • B$x$
   • C$x^6$
   • D$-x^6$

   (c)

   Write $\dfrac{x^5\times x^{-2}}{x^{-4}}$ as a single power of $x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Do not use a calculator in this question. Give each answer as an integer or a fraction in its simplest form.

   (a)

   Find the value of $2^{-3}$.

   [1 mark]

   • A$-8$
   • B$-6$
   • C$\frac{1}{6}$
   • D$\frac{1}{8}$

   (b)

   Find the value of $5^0\times2^{-3}\times16$.

   [1 mark]

   • A$\frac{1}{128}$
   • B$2$
   • C$0$
   • D$-128$

   (c)

   Find the value of $\left(\dfrac{2}{3}\right)^{-2}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Maya is simplifying algebraic expressions. All the letters represent non-zero numbers.

   (a)

   Simplify $\dfrac{12a^5b^{-2}}{4a^2b}$. Write your answer with positive exponents only.

   [3 marks]

   (b)

   Maya's working is:
   $(2x^3)^2\times x^{-5}=2x^6\times x^{-5}=2x$
   (i) Identify her error.
   (ii) Write the correct simplification, showing each step.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A scientist models a bacteria culture. The culture doubles in number every hour. At time $t=0$ there are $1000$ bacteria, so after $t$ hours (where negative $t$ means hours before the start) the number of bacteria is $N=1000\times2^t$.

   (a)

   (i) Use the index laws to find $N$ when $t=-1$, $t=-2$ and $t=-3$.
   (ii) Describe the pattern in these values and write the rule for $2^{-n}$.
   (iii) Use $2^3\div2^3$ to show that $2^0=1$, and verify that the model gives $N=1000$ at $t=0$.

   [6 marks]

   (b)

   A second culture is modelled by $N=800\times3^t$, where it trebles every hour.
   (i) Find $N$ when $t=-2$, and explain why you should round your answer to a whole number.
   (ii) Use a calculator to find $N$ when $t=12$.
   (iii) Comment on whether the model is likely to stay accurate for large values of $t$, giving a reason.

   [6 marks]

   Total for question 4: 12 marks

End of questions