Rationalising denominatorsIB MYP Maths Extended: Subtopic test

1. 1
   Lena is rationalising the denominators of fractions without a calculator.

   (a)

   Write $\frac{6}{\sqrt3}$ with a rational denominator, in its simplest form.

   [1 mark]

   • A$2\sqrt3$
   • B$6\sqrt3$
   • C$\frac{2}{\sqrt3}$
   • D$\frac{2\sqrt3}{3}$

   (b)

   Which expression should the numerator and denominator of $\frac{1}{3+\sqrt2}$ be multiplied by to give a rational denominator?

   [1 mark]

   • A$3+\sqrt2$
   • B$\sqrt2$
   • C$3-\sqrt2$
   • D$-3-\sqrt2$

   (c)

   Rationalise the denominator of $\frac{4}{\sqrt8}$ and simplify your answer.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Tomas is simplifying expressions that contain surds, without a calculator.

   (a)

   Write $\frac{2}{4-\sqrt3}$ with a rational denominator.

   [1 mark]

   • A$\frac{8-2\sqrt3}{13}$
   • B$\frac{8+2\sqrt3}{13}$
   • C$\frac{8+2\sqrt3}{19}$
   • D$8+2\sqrt3$

   (b)

   Simplify $\sqrt{18}+\sqrt{8}$.

   [1 mark]

   • A$\sqrt{26}$
   • B$13\sqrt2$
   • C$6\sqrt2$
   • D$5\sqrt2$

   (c)

   Simplify $\sqrt{12}\times\sqrt6$, giving your answer in the form $a\sqrt b$ where $b$ is as small as possible.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangular banner has area $12$ m$^2$. Give all answers as exact values with rational denominators.

   (a)

   The length of the banner is $\sqrt6$ m. Find its width.

   [3 marks]

   (b)

   A different banner has the same area and length $(3+\sqrt3)$ m. Find its width in the form $a+b\sqrt3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Ibrahim investigates the expressions $\frac{1}{\sqrt2+\sqrt1}$, $\frac{1}{\sqrt3+\sqrt2}$ and $\frac{1}{\sqrt4+\sqrt3}$.

   (a)

   (i) Rationalise the denominators of the first two expressions.
   (ii) Describe the pattern, write a general rule for $\frac{1}{\sqrt{n+1}+\sqrt n}$, and use it to predict the simplified form of $\frac{1}{\sqrt{10}+\sqrt9}$.

   [6 marks]

   (b)

   (i) Prove that $\frac{1}{\sqrt{n+1}+\sqrt n}=\sqrt{n+1}-\sqrt n$ for $n\ge0$.
   (ii) Hence find the exact value of $\frac{1}{\sqrt2+\sqrt1}+\frac{1}{\sqrt3+\sqrt2}+\frac{1}{\sqrt4+\sqrt3}+\dots+\frac{1}{\sqrt9+\sqrt8}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions