Laws of indices and surdsEdexcel International A Level Maths: Subtopic test

1. 1
   The number $N=16^{\frac34}\times27^{-\frac23}$.

   (a)

   Find the value of $16^{\frac34}$.

   [1 mark]

   • A$12$
   • B$2$
   • C$8$
   • D$4096$

   (b)

   Find the value of $27^{-\frac23}$.

   [1 mark]

   • A$\frac19$
   • B$-9$
   • C$\frac1{18}$
   • D$9$

   (c)

   Hence find the exact value of $N^{-\frac12}$, giving your answer in the form $k\sqrt2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For $x>0$, $f(x)=\dfrac{(2x^3)^2\sqrt{x}}{8x^5}$.

   (a)

   Which of the following is $f(x)$ in its simplest form?

   [1 mark]

   • A$\frac14x^{\frac32}$
   • B$\frac12x^{\frac32}$
   • C$\frac12x^{\frac12}$
   • D$2x^{\frac32}$

   (b)

   Find the value of $f(4)$.

   [1 mark]

   • A$1$
   • B$8$
   • C$16$
   • D$4$

   (c)

   Solve $f(x)=108$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangle has area $(7+\sqrt5)$ cm$^2$ and length $(3-\sqrt5)$ cm.

   (a)

   Find the width of the rectangle, giving your answer in the form $\frac{p+q\sqrt5}{r}$, where $p$, $q$ and $r$ are integers.

   [3 marks]

   (b)

   Find the exact perimeter of the rectangle, giving your answer in the form $a+b\sqrt5$, where $a$ and $b$ are integers.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $k=\dfrac{\sqrt{75}+\sqrt{12}}{\sqrt3-1}$.

   (a)

   (i) Show that $\sqrt{75}+\sqrt{12}=7\sqrt3$.
   (ii) Hence express $k$ in the form $\frac{p+q\sqrt3}{2}$, where $p$ and $q$ are integers.
   (iii) Explain why $k$ is irrational.

   [6 marks]

   (b)

   Given that $k=\frac{21+7\sqrt3}{2}$, show that $\frac{k^2}{2+\sqrt3}$ is rational, and find its value.

   [6 marks]

   Total for question 4: 12 marks

End of questions