1.10 Rational exponentsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Let $p=8^{\frac23}$ and $q=4^{-\frac12}$.

   (a)

   Find the value of $p$.

   [1 mark]

   • A$\frac{16}{3}$
   • B$2$
   • C$4$
   • D$64$

   (b)

   Find the value of $q$.

   [1 mark]

   • A$\frac12$
   • B$-2$
   • C$-\frac12$
   • D$2$

   (c)

   Write $pq$ as a single power of $2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cube has volume $V$ cm$^3$ and edge length $s$ cm, so that $s=V^{\frac13}$.

   (a)

   The surface area of the cube is $A=6s^{2}$. Which expression gives $A$ in terms of $V$?

   [1 mark]

   • A$6V^{\frac32}$
   • B$6V^{\frac13}$
   • C$36V^{\frac23}$
   • D$6V^{\frac23}$

   (b)

   Find the surface area of a cube of volume $27\,000$ cm$^3$.

   [1 mark]

   • A$900$ cm$^2$
   • B$5400$ cm$^2$
   • C$32\,400$ cm$^2$
   • D$162\,000$ cm$^2$

   (c)

   A cube has surface area $150$ cm$^2$. Find its volume.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   For $x>0$, consider $A=\frac{x^{\frac32}\times x^{-\frac12}}{\sqrt{x}}$ and $B=\frac{3}{\sqrt[3]{x^{2}}}$.

   (a)

   Write $A$ in the form $x^{k}$, where $k\in\mathbb{Q}$.

   [3 marks]

   (b)

   Find the value of $x$ for which $A=B$. Give your answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The orbital period $T$ years of a planet whose mean distance from the Sun is $R$ astronomical units (AU) satisfies $T=R^{\frac32}$.

   (a)

   (i) Find $T$ when $R=9$.
   (ii) Show that $R=T^{\frac23}$.
   (iii) Find $R$ when $T=64$.

   [6 marks]

   (b)

   (i) Planet Y has a mean distance from the Sun four times that of planet X. Show that $T_Y=8T_X$.
   (ii) Planet Z has a period of $5$ years. Find its mean distance from the Sun in AU, giving your answer exactly and to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions