1.02b Surds: manipulation and rationalising denominators

In this question you must show detailed reasoning.

1. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
2. Solve the equation \((8p^6)^{\frac{1}{3}} = 8\). [3]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

1. Solve the equation $$x\sqrt{2} - \sqrt{18} = x$$ writing the answer as a surd in simplest form. [3]
2. Solve the equation $$4^{3x-2} = \frac{1}{2\sqrt{2}}$$ [3]

In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. Simplify $$\frac{\sqrt{32} + \sqrt{18}}{3 + \sqrt{2}}$$ giving your answer in the form \(b\sqrt{2} + c\), where \(b\) and \(c\) are integers. [5]

In this question you must show detailed reasoning. Express \(\frac{8 + \sqrt{7}}{2 + \sqrt{7}}\) in the form \(a + b\sqrt{7}\), where \(a\) and \(b\) are integers. [4]

This question requires detailed reasoning. Express \(\frac{3 + \sqrt{20}}{3 + \sqrt{5}}\) in the form \(a + b\sqrt{5}\). [4]

Express \(\frac{a^{\frac{1}{2}} - a^{\frac{2}{3}}}{a^{\frac{1}{3}} - a}\) in the form \(a^m + \sqrt{a^n}\), where \(m\) and \(n\) are integers and \(a \neq 0\) or 1. [4]

In this question you must show detailed reasoning. Simplify \(10 + 7\sqrt{5} + \frac{38}{1 - 2\sqrt{5}}\), giving your answer in the form \(a + b\sqrt{5}\). [3]

Do not use a calculator for this question

1. Find \(a\), given that \(a^3 = 64x^{12}y^3\). [2]
2. 1. Express \(\frac{81}{\sqrt{3}}\) in the form \(3^k\). [2]
   2. Express \(\frac{5 + \sqrt{3}}{5 - \sqrt{3}}\) in the form \(\frac{a + b\sqrt{3}}{c}\), where \(a\), \(b\) and \(c\) are integers. [3]

Simplify fully.

1. \(\sqrt{a^3} \times \sqrt{16a}\) [2]
2. \((4b^6)^{\frac{3}{2}}\) [2]