1.02b Surds: manipulation and rationalising denominators

Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,

1. find the value of \(a\) and the value of \(b\). [2]

Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,

1. find the value of \(c\) and the value of \(d\). [3]

Express each of the following in the form \(a\sqrt{5}\), where \(a\) is an integer.

1. \(4\sqrt{15} \times \sqrt{3}\) [2]
2. \(\frac{20}{\sqrt{5}}\) [1]
3. \(5^{\frac{3}{2}}\) [1]

Express each of the following in the form \(k\sqrt{3}\), where \(k\) is an integer.

1. \(\frac{6}{\sqrt{3}}\) [1]
2. \(10\sqrt{3} - 6\sqrt{27}\) [2]
3. \(3^{\frac{3}{2}}\) [2]

Simplify \((3 + \sqrt{2})(3 - \sqrt{2})\). Express \(\frac{1 + \sqrt{2}}{3 - \sqrt{2}}\) in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are rational. [5]

Make \(y\) the subject of the formula \(a = \frac{\sqrt{y} - 5}{c}\). [3]

\includegraphics{figure_13} Fig. 13 shows the circle with equation \((x - 4)^2 + (y - 2)^2 = 16\).

1. Write down the radius of the circle and the coordinates of its centre. [2]
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. Give your answers in surd form. [4]
3. Show that the point A \((4 + 2\sqrt{2}, 2 + 2\sqrt{2})\) lies on the circle and mark point A on the copy of Fig. 13. Sketch the tangent to the circle at A and the other tangent that is parallel to it. Find the equations of both these tangents. [7]

Express \(\sqrt{50} + 3\sqrt{8}\) in the form \(k\sqrt{2}\). [3]

Express $$\frac{2}{3\sqrt{5} + 7}$$ in the form \(a + b\sqrt{5}\) where \(a\) and \(b\) are rational. [3]

\includegraphics{figure_1} Figure 1 shows the rectangles \(ABCD\) and \(EFGH\) which are similar. Given that \(AB = (3 - \sqrt{5})\) cm, \(AD = \sqrt{5}\) cm and \(EF = (1 + \sqrt{5})\) cm, find the length \(EH\) in cm, giving your answer in the form \(a + b\sqrt{5}\) where \(a\) and \(b\) are integers. [6]

\(\text{f}(x) = (2 - \sqrt{x})^2, \quad x > 0\).

1. Solve the equation \(\text{f}(x) = 0\). [2]
2. Find \(\text{f}(3)\), giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [2]
3. Find $$\int \text{f}(x) \, dx.$$ [4]

Express \(\sqrt{22.5}\) in the form \(k\sqrt{10}\). [4]

Fig. 8 shows a right-angled triangle with base \(2x + 1\), height \(h\) and hypotenuse \(3x\). \includegraphics{figure_1}

1. Show that \(h^2 = 5x^2 - 4x - 1\). [2]
2. Given that \(h = \sqrt{7}\), find the value of \(x\), giving your answer in surd form. [3]