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

1. 1
   The number $N$ is given by $N=\left(\frac{8}{27}\right)^{-\frac{2}{3}}$.

   (a)

   Which is equal to $\left(\frac{8}{27}\right)^{\frac{1}{3}}$?

   [1 mark]

   • A$\frac{2}{3}$
   • B$\frac{3}{2}$
   • C$\frac{2}{9}$
   • D$\frac{8}{9}$

   (b)

   What is the value of $N$?

   [1 mark]

   • A$\frac{4}{9}$
   • B$\frac{3}{2}$
   • C$\frac{9}{4}$
   • D$\frac{9}{2}$

   (c)

   Find the exact value of $N^{-\frac{1}{2}}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   It is given that $2^{x}=a$, where $a>0$.

   (a)

   Which expression is equal to $2^{x+3}$?

   [1 mark]

   • A$a+3$
   • B$3a$
   • C$a^3$
   • D$8a$

   (b)

   Which expression is equal to $\frac{1}{\sqrt{2^{x}}}$?

   [1 mark]

   • A$-\frac{a}{2}$
   • B$a^{-\frac12}$
   • C$\sqrt{a}$
   • D$\frac{1}{2a}$

   (c)

   Express $2^{3x-1}$ in terms of $a$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangle has width $(3+\sqrt{5})$ cm and area $(11+5\sqrt{5})$ cm$^2$.

   (a)

   Show that the length of the rectangle is $(2+\sqrt{5})$ cm.

   [3 marks]

   (b)

   The perimeter of the rectangle is $(10+4\sqrt{5})$ cm. Find the exact value of $\frac{\text{area}}{\text{perimeter}}$ in the form $a+b\sqrt{5}$, where $a$ and $b$ are rational numbers.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $x$ and $y$ be positive real numbers. No calculator may be used.

   (a)

   (i) Show that $\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}=\frac{x+2\sqrt{xy}+y}{x-y}$, where $x\neq y$.
   (ii) Hence find $\frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}$ in the form $a+b\sqrt{21}$.

   [6 marks]

   (b)

   Given that $\sqrt{x}-\sqrt{y}=1$ and $x-y=5$, find the values of $x$ and $y$.

   [6 marks]

   Total for question 4: 12 marks

End of questions