Rationalising the denominator questions with answers

Questions

1. Question 1A root in the denominator

   Rationalise the denominator and simplify $\dfrac{10}{\sqrt{3}}$.

   Show the worked solution

   1. Step 1: Multiply the top and the bottom by the surd in the denominator$\dfrac{10}{\sqrt{3}} = \dfrac{10 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}$$= \dfrac{10\sqrt{3}}{3}$
2. Question 2A root in the denominator

   Rationalise the denominator and simplify $\dfrac{5}{\sqrt{3}}$.

   Show the worked solution

   1. Step 1: Multiply the top and the bottom by the surd in the denominator$\dfrac{5}{\sqrt{3}} = \dfrac{5 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}$$= \dfrac{5\sqrt{3}}{3}$
3. Question 3Use the conjugate

   Express $\dfrac{20}{4 + \sqrt{6}}$ in the form $a + b\sqrt{6}$, where $a$ and $b$ are rational numbers.

   Show the worked solution

   1. Step 1: Multiply the top and the bottom by the conjugate of the bottom (change the sign between its terms)$\dfrac{20}{4 + \sqrt{6}} = \dfrac{(20)(4 - \sqrt{6})}{(4 + \sqrt{6})(4 - \sqrt{6})}$
   2. Step 2: The bottom is a difference of two squares: (p + q)(p − q) = p² − q²$\mathstrut 4^2 - (\sqrt{6})^2$$\text{Bottom} = 10$
   3. Step 3: Multiply out the top and write it over the bottom$\dfrac{20}{4 + \sqrt{6}} = \dfrac{80 - 20\sqrt{6}}{10}$
   4. Step 4: Simplify the fraction$= 8 - 2\sqrt{6}$
4. Question 4Use the conjugate

   Express $\dfrac{1}{4 + \sqrt{3}}$ in the form $a + b\sqrt{3}$, where $a$ and $b$ are rational numbers.

   Show the worked solution

   1. Step 1: Multiply the top and the bottom by the conjugate of the bottom (change the sign between its terms)$\dfrac{1}{4 + \sqrt{3}} = \dfrac{(1)(4 - \sqrt{3})}{(4 + \sqrt{3})(4 - \sqrt{3})}$
   2. Step 2: The bottom is a difference of two squares: (p + q)(p − q) = p² − q²$\mathstrut 4^2 - (\sqrt{3})^2$$\text{Bottom} = 13$
   3. Step 3: Multiply out the top and write it over the bottom$\dfrac{1}{4 + \sqrt{3}} = \dfrac{4 - \sqrt{3}}{13}$
5. Question 5Surds on the top and the bottom

   Express $\dfrac{1 + \sqrt{2}}{1 - \sqrt{2}}$ in the form $\dfrac{a + b\sqrt{2}}{c}$, where $a$, $b$ and $c$ are integers.

   Show the worked solution

   1. Step 1: Multiply the top and the bottom by the conjugate of the bottom (change the sign between its terms)$\dfrac{1 + \sqrt{2}}{1 - \sqrt{2}} = \dfrac{(1 + \sqrt{2})(1 + \sqrt{2})}{(1 - \sqrt{2})(1 + \sqrt{2})}$
   2. Step 2: The bottom is a difference of two squares: (p + q)(p − q) = p² − q²$\mathstrut 1^2 - (\sqrt{2})^2$$\text{Bottom} = -1$
   3. Step 3: Expand the top (every term by every term) and write it over the bottom$\dfrac{1 + \sqrt{2}}{1 - \sqrt{2}} = \dfrac{3 + 2\sqrt{2}}{-1}$
   4. Step 4: Simplify the fraction$= -3 - 2\sqrt{2}$
6. Question 6Surds on the top and the bottom

   Express $\dfrac{1 + 2\sqrt{5}}{3 - \sqrt{5}}$ in the form $\dfrac{a + b\sqrt{5}}{c}$, where $a$, $b$ and $c$ are integers.

   Show the worked solution

   1. Step 1: Multiply the top and the bottom by the conjugate of the bottom (change the sign between its terms)$\dfrac{1 + 2\sqrt{5}}{3 - \sqrt{5}} = \dfrac{(1 + 2\sqrt{5})(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})}$
   2. Step 2: The bottom is a difference of two squares: (p + q)(p − q) = p² − q²$\mathstrut 3^2 - (\sqrt{5})^2$$\text{Bottom} = 4$
   3. Step 3: Expand the top (every term by every term) and write it over the bottom$\dfrac{1 + 2\sqrt{5}}{3 - \sqrt{5}} = \dfrac{13 + 7\sqrt{5}}{4}$