Fractional exponentsIB MYP Maths Extended: Subtopic test

1. 1
   Mia is exploring powers of $64$ without a calculator.

   (a)

   Find the value of $64^{\frac{1}{3}}$.

   [1 mark]

   • A$\frac{64}{3}$
   • B$4$
   • C$8$
   • D$32$

   (b)

   Find the value of $64^{\frac{2}{3}}$.

   [1 mark]

   • A$\frac{128}{3}$
   • B$4$
   • C$128$
   • D$16$

   (c)

   Evaluate $64^{-\frac{5}{6}}$, giving your answer as a fraction.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Kofi is simplifying algebraic expressions in which $x>0$.

   (a)

   Simplify $\left(x^{\frac12}\right)^{4}$.

   [1 mark]

   • A$x^{2}$
   • B$x^{\frac18}$
   • C$x^{\frac92}$
   • D$x^{\frac72}$

   (b)

   Simplify $x^{\frac23}\times x^{\frac16}$.

   [1 mark]

   • A$x^{\frac19}$
   • B$x^{\frac13}$
   • C$x^{\frac56}$
   • D$x^{\frac12}$

   (c)

   Simplify $\sqrt[3]{x}\div x^{\frac34}$, giving your answer as a single power of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The orbital period $T$ (in Earth years) of a planet is modelled by $T=r^{\frac{3}{2}}$, where $r$ is the planet's average distance from its star in astronomical units (AU).

   (a)

   A dwarf planet has $r=25$. Find its orbital period $T$.

   [3 marks]

   (b)

   A different planet has period $T=8$ years. Find $r$, showing how you rearrange the formula.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Biologists model the daily energy use $R$ (in kilocalories) of a mammal of mass $m$ kg by $R=70m^{\frac{3}{4}}$.

   (a)

   (i) Find $R$ for a mammal of mass $16$ kg.
   (ii) Find $R$ for a mammal of mass $81$ kg.
   (iii) Show that when the mass is multiplied by $16$, the value of $R$ is multiplied by $8$.

   [6 marks]

   (b)

   A zoo vet uses the model for an elephant of mass $4096$ kg and a rat of mass $\frac{1}{16}$ kg. (i) Find $R$ for each animal. (ii) The elephant is $65\,536$ times heavier than the rat. Justify whether the elephant needs $65\,536$ times as much energy each day.

   [6 marks]

   Total for question 4: 12 marks

End of questions