Indices and Surds

2 Express \(\frac { 3 + \sqrt { 20 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).

5. (a) Write \(\sqrt { 45 }\) in the form \(a \sqrt { 5 }\), where \(a\) is an integer.
(b) Express \(\frac { 2 ( 3 + \sqrt { 5 } ) } { ( 3 - \sqrt { 5 } ) }\) in the form \(b + c \sqrt { 5 }\), where \(b\) and \(c\) are integers.
\section*{6.} \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the points \(( 0,3 )\) and \(( 4,0 )\) and touches the \(x\)-axis at the point \(( 1,0 )\). On separate diagrams sketch the curve with equation

3 In this question you must show detailed reasoning.
Find the value of \(k\) such that \(\frac { 1 } { \sqrt { 5 } + \sqrt { 6 } } + \frac { 1 } { \sqrt { 6 } + \sqrt { 7 } } = \frac { k } { \sqrt { 5 } + \sqrt { 7 } }\).

Evaluate \(49^{\frac{1}{2}} + 8^{\frac{1}{3}}\). [3]

State the value of each of the following.

1. \(2^{-3}\) [1]
2. \(9^0\) [1]

2 A civilisation which works in base 5 sends out the first 6 digits of \(\pi\) as 3.032 32. Convert this to base 10.

Factorise completely \(x - 4 x ^ { 3 }\)

1 Express as a single power of \(a\) $$\frac { a ^ { 2 } } { \sqrt { a } }$$ where \(a \neq 0\) Circle your answer. \(a ^ { 1 }\) \(a ^ { \frac { 3 } { 2 } }\) \(a ^ { \frac { 5 } { 2 } }\) \(a ^ { 4 }\)

Given that \(x > 0\) and \(x \neq 25\), fully simplify $$\frac{10 + 5x - 2x^{\frac{1}{2}} - x^{\frac{3}{2}}}{5 - \sqrt{x}}$$ Fully justify your answer. [4 marks]

Simplify \((3 + \sqrt{5})(3 - \sqrt{5})\). [2]

Simplify \(( 3 + \sqrt { } 5 ) ( 3 - \sqrt { } 5 )\). \includegraphics[max width=\textwidth, alt={}, center]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-02_108_93_2614_1786}

2. (a) Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
(b) Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.

Make \(a\) the subject of \(3(a + 4) = ac + 5f\). [4]

1 Solve the equations

1. \(x ^ { \frac { 1 } { 3 } } = 2\),
2. \(10 ^ { \prime } = 1\),
3. \(\left( y ^ { - 2 } \right) ^ { 2 } = \frac { 1 } { 81 }\).

6. $$f ( x ) = x ^ { \frac { 3 } { 2 } } - 8 x ^ { - \frac { 1 } { 2 } }$$

1. Evaluate \(\mathrm { f } ( 3 )\), giving your answer in its simplest form with a rational denominator.
2. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(k \sqrt { 2 }\).

2. Given that \(32 \sqrt { } 2 = 2 ^ { a }\), find the value of \(a\).

2. Given that \(32 \sqrt { } 2 = 2 ^ { a }\), find the value of \(a\).

1. Find the value of \(a\) and the value of \(b\) for which \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } \equiv 2 ^ { a x + b }\)
2. Hence solve the equation \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } = 2 \sqrt { 2 }\)

1 Show that \(\frac { 31 } { 6 - \sqrt { 5 } } = 6 + \sqrt { 5 }\).

5 Show that the distance between the points \(( 5,2 )\) and \(( 11 , - 1 )\) is \(a \sqrt { b }\), where \(a\) and \(b\) are integers to be determined.

Show that $$\frac{3 + \sqrt{8n}}{1 + \sqrt{2n}}$$ can be written as $$\frac{4n - 3 + \sqrt{2n}}{2n - 1}$$ where \(n\) is a positive integer. [4 marks]

Find the value of \(y\) such that $$4^{y + 3} = 8.$$ [3]

1. (i) Simplify

$$\sqrt { 48 } - \frac { 6 } { \sqrt { 3 } }$$ Write your answer in the form \(a \sqrt { 3 }\), where \(a\) is an integer to be found.
(ii) Solve the equation $$3 ^ { 6 x - 3 } = 81$$ Write your answer as a rational number.

1 Find the set of values of \(x\) for which \(3 \left( 2 ^ { 3 x + 1 } \right) < 8\). Give your answer in a simplified exact form.

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

1. Write

$$\sqrt { } ( 75 ) - \sqrt { } ( 27 )$$ in the form \(k \sqrt { } x\), where \(k\) and \(x\) are integers.

1 Express \(\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }\) in the form \(k \sqrt { 5 }\), where \(k\) is an integer.

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

1. (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.
[0pt]

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.

8. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$

1. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
2. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\). END

1. Given

$$\frac { 3 ^ { x } } { 3 ^ { 4 y } } = 27 \sqrt { 3 }$$ find \(y\) as a simplified function of \(x\).

Express $$5 - \frac{\sqrt[3]{x}}{x^2}$$ in the form $$5x^p - x^q$$ where \(p\) and \(q\) are constants. [2 marks]

Express \(\frac{\sqrt{3} + 3\sqrt{5}}{\sqrt{5} - \sqrt{3}}\) in the form \(a + b\sqrt{c}\), where \(a\) and \(b\) are integers. Fully justify your answer. [4 marks]

1. $$f ( x ) = ( \sqrt { x } + 3 ) ^ { 2 } + ( 1 - 3 \sqrt { x } ) ^ { 2 }$$ Show that \(\mathrm { f } ( x )\) can be written in the form \(a x + b\) where \(a\) and \(b\) are integers to be found.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
   1. Expand and simplify
   $$\left( r - \frac { 1 } { r } \right) ^ { 2 }$$
2. Express \(\frac { 1 } { 3 + 2 \sqrt { 2 } }\) in the form \(p + q \sqrt { 2 }\) where \(p\) and \(q\) are integers.
3. Use the results of parts (a) and (b), or otherwise, to show that $$\sqrt { 3 + 2 \sqrt { 2 } } - \frac { 1 } { \sqrt { 3 + 2 \sqrt { 2 } } } = 2$$

1. Given that

$$\left( 3 p q ^ { 2 } \right) ^ { 4 } \times 2 p \sqrt { q ^ { 8 } } \equiv a p ^ { b } q ^ { c }$$ find the values of the constants \(a , b\) and \(c\).