Rationalize denominator simple

1 Express \(\frac { 15 + \sqrt { 3 } } { 3 - \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.

2 Express \(\frac { 8 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.

3

1. Express \(\frac { 12 } { 3 + \sqrt { 5 } }\) in the form \(a - b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.
2. Express \(\sqrt { 18 } - \sqrt { 2 }\) in simplified surd form.

1 Express \(\frac { 8 } { \sqrt { 3 } - 1 }\) in the form \(a \sqrt { 3 } + b\), where \(a\) and \(b\) are integers.

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

5

1. Find the value of \(144 ^ { - \frac { 1 } { 2 } }\).
2. Simplify \(\frac { 1 } { 5 + \sqrt { 7 } } + \frac { 4 } { 5 - \sqrt { 7 } }\). Give your answer in the form \(\frac { a + b \sqrt { 7 } } { c }\).

7

1. Express \(\sqrt { 48 } + \sqrt { 75 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
2. Simplify \(\frac { 7 + 2 \sqrt { 5 } } { 7 + \sqrt { 5 } }\), expressing your answer in the form \(\frac { a + b \sqrt { 5 } } { c }\), where \(a , b\) and \(c\) are integers.

5

1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
2. Express \(\frac { 5 + 2 \sqrt { 3 } } { 4 - \sqrt { 3 } }\) in the form \(c + d \sqrt { 3 }\), where \(c\) and \(d\) are integers.

1 Write \(\frac { 8 } { 3 - \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers to be found.

3 Given that \(k\) is an integer, express \(\frac { 3 \sqrt { 2 } - k } { \sqrt { 8 } + 1 }\) in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational expressions in terms of \(k\).

1 Rationalise the denominator of the fraction \(\frac { 2 + \sqrt { n } } { 3 + \sqrt { n } }\), where \(n\) is a positive integer.

4 In this question you must show detailed reasoning.
Express \(\frac { 1 + 4 \sqrt { 3 } } { 2 + \sqrt { 3 } }\) in the form \(\mathrm { a } + \mathrm { b } \sqrt { 3 }\), where \(a\) and \(b\) are integers to be determined.

2 Express \(\frac { a + \sqrt { 2 } } { 3 - \sqrt { 2 } }\) in the form \(\mathrm { p } + \mathrm { q } \sqrt { 2 }\), giving \(p\) and \(q\) in terms of \(a\).

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

3

1. Express \(\frac { 7 + \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
2. Express \(\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }\) in the form \(k \sqrt { 5 }\), where \(k\) is an integer.

2

1. Simplify \(( 3 \sqrt { 3 } ) ^ { 2 }\).
2. Express \(\frac { 4 \sqrt { 3 } + 3 \sqrt { 7 } } { 3 \sqrt { 3 } + \sqrt { 7 } }\) in the form \(\frac { m + \sqrt { 21 } } { n }\), where \(m\) and \(n\) are integers.

3

1. 1. Express \(\sqrt { 18 }\) in the form \(k \sqrt { 2 }\), where \(k\) is an integer.
   2. Simplify \(\frac { \sqrt { 8 } } { \sqrt { 18 } + \sqrt { 32 } }\).
2. Express \(\frac { 7 \sqrt { 2 } - \sqrt { 3 } } { 2 \sqrt { 2 } - \sqrt { 3 } }\) in the form \(m + \sqrt { n }\), where \(m\) and \(n\) are integers.

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 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

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 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 \(5 \sqrt { 8 } + \frac { 6 } { \sqrt { 2 } }\) 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

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 \(x\).