Solve equations with surds

2. Without using your calculator, solve $$x \sqrt { 27 } + 21 = \frac { 6 x } { \sqrt { 3 } }$$ Write your answer in the form \(a \sqrt { b }\) where \(a\) and \(b\) are integers. You must show all stages of your working.

5. Solve the equation $$10 + x \sqrt { 8 } = \frac { 6 x } { \sqrt { 2 } }$$ Give your answer in the form \(a \sqrt { } b\) where \(a\) and \(b\) are integers.

1. (i) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
   (ii) Find the value of \(x\) such that

$$\frac { 1 + x } { x } = \sqrt { 3 } ,$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.

1. 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.
2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$

2

1. 1. Express \(\sqrt { 48 }\) in the form \(n \sqrt { 3 }\), where \(n\) is an integer.
   2. Solve the equation $$x \sqrt { 12 } = 7 \sqrt { 3 } - \sqrt { 48 }$$ giving your answer in its simplest form.
2. Express \(\frac { 11 \sqrt { 3 } + 2 \sqrt { 5 } } { 2 \sqrt { 3 } + \sqrt { 5 } }\) in the form \(m - \sqrt { 15 }\), where \(m\) is an integer.

4. (a) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
(b) Find the value of \(x\) such that $$\frac { 1 + x } { x } = \sqrt { 3 } ,$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.

1 In this question you must show detailed reasoning. Solve the equation \(x ( 3 - \sqrt { 5 } ) = 24\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.

2 In this question you must show detailed reasoning. Solve the equation \(x \sqrt { 5 } + 32 = x \sqrt { 45 } + 2 x\). Give your answer in the form \(a \sqrt { 5 } + b\), where \(a\) and \(b\) are integers to be determined.

3

1. Express \(\frac { \sqrt { 5 } + 3 } { \sqrt { 5 } - 2 }\) in the form \(p \sqrt { 5 } + q\), where \(p\) and \(q\) are integers.
2. 1. Express \(\sqrt { 45 }\) in the form \(n \sqrt { 5 }\), where \(n\) is an integer.
   2. Solve the equation $$x \sqrt { 20 } = 7 \sqrt { 5 } - \sqrt { 45 }$$ giving your answer in its simplest form.

1 Solve the equation $$x \sqrt { 32 } - \sqrt { 24 } = ( 3 \sqrt { 3 } - 5 ) ( \sqrt { 6 } + x \sqrt { 2 } )$$

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]

In the question you must show detailed reasoning Solve the equation below, giving your answer in the simplest form $$x\sqrt{32} - \sqrt{24} = (3\sqrt{3} - 5)(\sqrt{6} + x\sqrt{2})$$ [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]