Rationalize denominator simple

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. Simplify $$\frac { 5 - 2 \sqrt { 3 } } { \sqrt { 3 } - 1 }$$ giving your answer in the form \(p + q \sqrt { } 3\), where \(p\) and \(q\) are rational numbers.

6. (a) Expand and simplify \(( 4 + \sqrt { 3 } ) ( 4 - \sqrt { 3 } )\).
(b) Express \(\frac { 26 } { 4 + \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.

1. Simplify

$$\frac { 7 + \sqrt { 5 } } { \sqrt { 5 } - 1 }$$ giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.

Simplify $$\frac { 5 - \sqrt { 3 } } { 2 + \sqrt { 3 } } ,$$ giving your answer in the form \(a + b \sqrt { } 3\), where \(a\) and \(b\) are integers.

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

1 Express \(\frac { 4 } { 3 - \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.

5

1. Simplify \(2 x ^ { \frac { 2 } { 3 } } \times 3 x ^ { - 1 }\).
2. Express \(2 ^ { 40 } \times 4 ^ { 30 }\) in the form \(2 ^ { n }\).
3. Express \(\frac { 26 } { 4 - \sqrt { } 3 }\) in the form \(a + b \sqrt { } 3\).

8

1. Write \(\sqrt { 48 } + \sqrt { 3 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
2. Simplify \(\frac { 1 } { 5 + \sqrt { 2 } } + \frac { 1 } { 5 - \sqrt { 2 } }\).

8

1. Simplify \(\sqrt { 98 } - \sqrt { 50 }\).
2. Express \(\frac { 6 \sqrt { 5 } } { 2 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.

7

1. Express \(\frac { 1 } { 5 + \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
2. Expand and simplify \(( 3 - 2 \sqrt { 7 } ) ^ { 2 }\).

8 Find the values of \(a\) and \(b\) for which \(\frac { 4 } { ( 2 \sqrt { 3 } - 1 ) } = a + b \sqrt { 3 }\).

1. Express

$$\frac { 2 } { 3 \sqrt { 5 } + 7 }$$ in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational.

4

1. Express \(125 \sqrt { 5 }\) in the form \(5 ^ { k }\).
2. Simplify \(10 + 7 \sqrt { 5 } + \frac { 38 } { 1 - 2 \sqrt { 5 } }\), giving your answer in the form \(a + b \sqrt { 5 }\).

5

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.

9

1. Express \(\sqrt { 48 } + \sqrt { 27 }\) in the form \(a \sqrt { 3 }\).
2. Simplify \(\frac { 5 \sqrt { 2 } } { 3 - \sqrt { 2 } }\). Give your answer in the form \(\frac { b + c \sqrt { 2 } } { d }\).

11

1. Express \(\sqrt { 75 } + \sqrt { 48 }\) in the form \(a \sqrt { 3 }\).
2. Express \(\frac { 14 } { 3 - \sqrt { 2 } }\) in the form \(b + c \sqrt { d }\).

12

1. Express \(\frac { 1 } { 5 + \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
2. Expand and simplify \(( 3 - 2 \sqrt { 7 } ) ^ { 2 }\).

15

1. Simplify \(\sqrt { 98 } \quad \sqrt { 50 }\).
2. Express \(\frac { 6 \sqrt { 5 } } { 2 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.

17

1. Simplify \(5 \sqrt { 8 } + 4 \sqrt { 50 }\). Express your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
2. Express \(\frac { \sqrt { 3 } } { 6 \sqrt { 3 } }\) in the form \(p + q \sqrt { 3 }\), where \(p\) and \(q\) are rational.

6

1. Write \(\sqrt { } 80\) in the form \(c \sqrt { } 5\), where \(c\) is a positive constant. A rectangle \(R\) has a length of ( \(1 + \sqrt { } 5\) ) cm and an area of \(\sqrt { 80 } \mathrm {~cm} ^ { 2 }\).
2. Calculate the width of \(R\) in cm . Express your answer in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers to be found.