1.02a Indices: laws of indices for rational exponents

4 Solve the equation \(2 y ^ { \frac { 1 } { 2 } } - 7 y ^ { \frac { 1 } { 4 } } + 3 = 0\).

5 Express the following in the form \(2 ^ { p }\).

1. \(\left( 2 ^ { 5 } \div 2 ^ { 7 } \right) ^ { 3 }\)
2. \(5 \times 4 ^ { \frac { 2 } { 3 } } + 3 \times 16 ^ { \frac { 1 } { 3 } }\)

6 Find the value of each of the following, giving each answer as an integer or fraction as appropriate.

1. \(25 ^ { \frac { 3 } { 2 } }\)
2. \(\left( \frac { 7 } { 3 } \right) ^ { - 2 }\)

9

1. Simplify \(3 a ^ { 3 } b \times 4 ( a b ) ^ { 2 }\).
2. Factorise \(x ^ { 2 } - 4\) and \(x ^ { 2 } - 5 x + 6\). Hence express \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } - 5 x + 6 }\) as a fraction in its simplest form.

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 \(\frac { a + b \sqrt { 7 } } { c }\).

2

1. Write down the value of each of the following.
   (A) \(4 ^ { - 2 }\) (B) \(9 ^ { 0 }\)
2. Find the value of \(\left( \frac { 64 } { 125 } \right) ^ { \frac { 4 } { 3 } }\).

3 Simplify \(\frac { \left( 3 x y ^ { 4 } \right) ^ { 3 } } { 6 x ^ { 5 } y ^ { 2 } }\).

5 The volume \(V\) of a cone with base radius \(r\) and slant height \(l\) is given by the formula $$V = \frac { 1 } { 3 } \pi r ^ { 2 } \sqrt { l ^ { 2 } - r ^ { 2 } }$$ Rearrange this formula to make \(l\) the subject.

7

1. Express \(\frac { 81 } { \sqrt { 3 } }\) in the form \(3 ^ { k }\).
2. Express \(\frac { 5 + \sqrt { 3 } } { 5 - \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.

2

1. Evaluate \(9 ^ { - \frac { 1 } { 2 } }\).
2. Simplify \(\frac { \left( 4 x ^ { 4 } \right) ^ { 3 } y ^ { 2 } } { 2 x ^ { 2 } y ^ { 5 } }\).

1 Find the value of each of the following.

1. \(\left( \frac { 5 } { 3 } \right) ^ { - 2 }\)
2. \(81 ^ { \frac { 3 } { 4 } }\)

2 Simplify \(\frac { \left( 4 x ^ { 5 } y \right) ^ { 3 } } { \left( 2 x y ^ { 2 } \right) \times \left( 8 x ^ { 10 } y ^ { 4 } \right) }\).

1

1. Evaluate \(\left( \frac { 1 } { 27 } \right) ^ { \frac { 2 } { 3 } }\).
2. Simplify \(\frac { \left( 4 a ^ { 2 } c \right) ^ { 3 } } { 32 a ^ { 4 } c ^ { 7 } }\).

1 Find the value of each of the following.

1. \(3 ^ { 0 }\)
2. \(9 ^ { \frac { 3 } { 2 } }\)
3. \(\left( \frac { 4 } { 5 } \right) ^ { - 2 }\)

3

1. Solve the inequality \(\frac { 1 - 2 x } { 4 } > 3\).
2. Simplify \(\left( 5 c ^ { 2 } d \right) ^ { 3 } \times \frac { 2 c ^ { 4 } } { d ^ { 5 } }\).

9 Fig. 9 shows the curve with equation \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\). It has an asymptote \(x = a\) and turning point P . \begin{figure}[h] \end{figure}

1. Write down the value of \(a\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 3 x ^ { 2 } } { 3 y ^ { 2 } ( 2 x - 1 ) ^ { 2 } }\). Hence find the coordinates of the turning point P , giving the \(y\)-coordinate to 3 significant figures.
3. Show that the substitution \(u = 2 x - 1\) transforms \(\int \frac { x } { \sqrt [ 3 ] { 2 x - 1 } } \mathrm {~d} x\) to \(\frac { 1 } { 4 } \int \left( u ^ { \frac { 2 } { 3 } } + u ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } u\). Hence find the exact area of the region enclosed by the curve \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4.5\).

2 It is given that \(\mathrm { f } ( n ) = 2 ^ { 3 n } + 8 ^ { n - 1 }\). By simplifying \(\mathrm { f } ( k ) + \mathrm { f } ( k + 1 )\), or otherwise, prove by mathematical induction that \(\mathrm { f } ( n )\) is divisible by 9 for every positive integer \(n\).

2 Simplify fully.

1. \(\sqrt { 12 a } \times \sqrt { 3 a ^ { 5 } }\)
2. \(\left( 64 b ^ { 3 } \right) ^ { \frac { 1 } { 3 } } \times \left( 4 b ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\)
3. \(7 \times 9 ^ { 3 c } - 4 \times 27 ^ { 2 c }\)

4 In this question you must show detailed reasoning.
The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 11 x + 6\).

1. Use the factor theorem to show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in fully factorised form.
3. Hence solve the equation \(2 \times 8 ^ { y } - 3 \times 4 ^ { y } - 11 \times 2 ^ { y } + 6 = 0\).

1 Simplify fully.

1. \(\sqrt { a ^ { 3 } } \times \sqrt { 16 a }\)
2. \(\quad \left( 4 b ^ { 6 } \right) ^ { \frac { 5 } { 2 } }\)

1. In this question you should show all stages of your working.

Solutions relying on calculator technology are not acceptable.
Given $$\frac { 9 ^ { x - 1 } } { 3 ^ { y + 2 } } = 81$$ express \(y\) in terms of \(x\), writing your answer in simplest form.

1. The first 3 terms of a geometric sequence are

$$3 ^ { 4 k - 5 } \quad 9 ^ { 7 - 2 k } \quad 3 ^ { 2 ( k - 1 ) }$$ where \(k\) is a constant.

1. Using algebra and making your reasoning clear, prove that \(k = \frac { 5 } { 2 }\)
2. Hence find the sum to infinity of the geometric sequence.

1. Given

$$2 ^ { x } \times 4 ^ { y } = \frac { 1 } { 2 \sqrt { 2 } }$$ express \(y\) as a function of \(x\).

11 In this question you must show detailed reasoning.

1. A student is asked to solve the inequality \(x ^ { \frac { 1 } { 2 } } < 4\). The student argues that \(x ^ { \frac { 1 } { 2 } } < 4 \Leftrightarrow x < 16\), so that the solution is \(\{ x : x < 16 \}\).
   Comment on the validity of the student's argument.
2. Solve the inequality \(\left( \frac { 1 } { 2 } \right) ^ { x } < 4\).
3. Show that the equation \(2 \log _ { 2 } ( x + 8 ) - \log _ { 2 } ( x + 6 ) = 3\) has only one root.