1.02b Surds: manipulation and rationalising denominators

5 Express each of the following in the form \(m + n \sqrt { 3 }\), where \(m\) and \(n\) are integers:

1. \(( \sqrt { 3 } + 1 ) ^ { 2 }\);
2. \(\frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).

4

1. Express \(( 4 \sqrt { 5 } - 1 ) ( \sqrt { 5 } + 3 )\) in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers.
2. Show that \(\frac { \sqrt { 75 } - \sqrt { 27 } } { \sqrt { 3 } }\) is an integer and find its value.

2

1. Express \(( 3 - \sqrt { 5 } ) ^ { 2 }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
2. Hence express \(\frac { ( 3 - \sqrt { 5 } ) ^ { 2 } } { 1 + \sqrt { 5 } }\) in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers.
   (4 marks)

2

1. 1. Express \(\sqrt { 48 }\) in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
   2. Simplify \(\frac { \sqrt { 48 } + 2 \sqrt { 27 } } { \sqrt { 12 } }\), giving your answer as an integer.
2. Express \(\frac { 1 - 5 \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.

1 Express \(\frac { 5 \sqrt { 3 } - 6 } { 2 \sqrt { 3 } + 3 }\) in the form \(m + n \sqrt { 3 }\), where \(m\) and \(n\) are integers.
(4 marks)

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.

2 A rectangle has length \(( 9 + 5 \sqrt { 3 } ) \mathrm { cm }\) and area \(( 15 + 7 \sqrt { 3 } ) \mathrm { cm } ^ { 2 }\).
Find the width of the rectangle, giving your answer in the form \(( m + n \sqrt { 3 } ) \mathrm { cm }\), where \(m\) and \(n\) are integers.
[0pt] [4 marks]

2 The point \(P\) has coordinates \(( \sqrt { 3 } , 2 \sqrt { 3 } )\) and the point \(Q\) has coordinates \(( \sqrt { 5 } , 4 \sqrt { 5 } )\). Show that the gradient of \(P Q\) can be expressed as \(n + \sqrt { 15 }\), stating the value of the integer \(n\).
[0pt] [5 marks]

2

1. Simplify \(( 3 \sqrt { 5 } ) ^ { 2 }\).
2. Express \(\frac { ( 3 \sqrt { 5 } ) ^ { 2 } + \sqrt { 5 } } { 7 + 3 \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
   [0pt] [4 marks]

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.

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

1. Evaluate 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 \(k \sqrt { 2 }\).

1. Find the integer \(n\) such that

$$4 \sqrt { 12 } - \sqrt { 75 } = \sqrt { n }$$

2. Express \(\sqrt { 22.5 }\) in the form \(k \sqrt { 10 }\).

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 \(3 x + 1 = 4 \sqrt { x }\).

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.

4 In this question you must show detailed reasoning. Determine the exact value of \(\frac { 1 } { \sqrt { 2 } + 1 } + \frac { 1 } { \sqrt { 3 } + \sqrt { 2 } } + \frac { 1 } { 2 + \sqrt { 3 } }\).

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.

3

1. Express \(5 \sqrt { 8 } + \frac { 6 } { \sqrt { 2 } }\) in the form \(n \sqrt { 2 }\), where \(n\) is an integer.
2. Express \(\frac { \sqrt { 2 } + 2 } { 3 \sqrt { 2 } - 4 }\) in the form \(c \sqrt { 2 } + d\), where \(c\) and \(d\) are integers.

2

1. Express \(\frac { \sqrt { 63 } } { 3 } + \frac { 14 } { \sqrt { 7 } }\) in the form \(n \sqrt { 7 }\), where \(n\) is an integer.
2. Express \(\frac { \sqrt { 7 } + 1 } { \sqrt { 7 } - 2 }\) in the form \(p \sqrt { 7 } + q\), where \(p\) and \(q\) are integers.

2 It is given that \(x = \sqrt { 3 }\) and \(y = \sqrt { 12 }\).
Find, in the simplest form, the value of:

1. \(x y\);
2. \(\frac { y } { x }\);
3. \(( x + y ) ^ { 2 }\).

2

1. Express \(\frac { 5 + \sqrt { 7 } } { 3 - \sqrt { 7 } }\) in the form \(m + n \sqrt { 7 }\), where \(m\) and \(n\) are integers.
2. The diagram shows a right-angled triangle. The hypotenuse has length \(2 \sqrt { 5 } \mathrm {~cm}\). The other two sides have lengths \(3 \sqrt { 2 } \mathrm {~cm}\) and \(x \mathrm {~cm}\). Find the value of \(x\).

4 For a small angle \(\theta\), where \(\theta\) is in radians, show that \(1 + \cos \theta - 3 \cos ^ { 2 } \theta \approx - 1 + \frac { 5 } { 2 } \theta ^ { 2 }\).

7 The expression $$\frac { 3 - \sqrt { } n } { 2 + \sqrt { } n }$$ can be written in the form \(a + b \sqrt { } n\), where \(a\) and \(b\) and \(n\) are rational but \(\sqrt { } n\) is irrational. Find expressions for \(a\) and \(b\) in terms of \(n\).

7

1. Given that \(n\) is a positive integer, express $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ as a single fraction not involving surds.
   7
2. Hence, deduce that $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ is a rational number for all positive integer values of \(n\)