1.02a Indices: laws of indices for rational exponents

Simplify fully

1. \(\left( 25 x ^ { 4 } \right) ^ { \frac { 1 } { 2 } }\),
2. \(\left( 25 x ^ { 4 } \right) ^ { - \frac { 3 } { 2 } }\).

5. (a) Write \(\frac { 2 \sqrt { } x + 3 } { x }\) in the form \(2 x ^ { p } + 3 x ^ { q }\) where \(p\) and \(q\) are constants. Given that \(y = 5 x - 7 + \frac { 2 \sqrt { } x + 3 } { x } , \quad x > 0\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.

7. (a) Show that \(\frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x }\) can be written as \(9 x ^ { - \frac { 1 } { 2 } } - 6 + x ^ { \frac { 1 } { 2 } }\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , x > 0\), and that \(y = \frac { 2 } { 3 }\) at \(x = 1\),
(b) find \(y\) in terms of \(x\).

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

9. $$f ( x ) = \frac { ( 3 - 4 \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , \quad x > 0$$

1. Show that \(\mathrm { f } ( x ) = 9 x ^ { - \frac { 1 } { 2 } } + A x ^ { \frac { 1 } { 2 } } + B\), where \(A\) and \(B\) are constants to be found.
2. Find \(\mathrm { f } ^ { \prime } ( x )\).
3. Evaluate \(\mathrm { f } ^ { \prime } ( 9 )\).

Find the value of

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

6. Given that \(\frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\) can be written in the form \(6 x ^ { p } + 3 x ^ { q }\),

1. write down the value of \(p\) and the value of \(q\). Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }\), and that \(y = 90\) when \(x = 4\),
2. find \(y\) in terms of \(x\), simplifying the coefficient of each term.

2. (a) Evaluate \(( 32 ) ^ { \frac { 3 } { 5 } }\), giving your answer as an integer.
(b) Simplify fully \(\left( \frac { 25 x ^ { 4 } } { 4 } \right) ^ { - \frac { 1 } { 2 } }\)

5. Solve

1. \(2 ^ { y } = 8\)
2. \(2 ^ { x } \times 4 ^ { x + 1 } = 8\)

3. (a) Find the value of \(8 ^ { \frac { 5 } { 3 } }\) (b) Simplify fully \(\frac { \left( 2 x ^ { \frac { 1 } { 2 } } \right) ^ { 3 } } { 4 x ^ { 2 } }\)

Express \(9 ^ { 3 x + 1 }\) in the form \(3 ^ { y }\), giving \(y\) in the form \(a x + b\), where \(a\) and \(b\) are constants.
(2)

6. (a) Given \(y = 2 ^ { x }\), show that $$2 ^ { 2 x + 1 } - 17 \left( 2 ^ { x } \right) + 8 = 0$$ can be written in the form $$2 y ^ { 2 } - 17 y + 8 = 0$$ (b) Hence solve $$2 ^ { 2 x + 1 } - 17 \left( 2 ^ { x } \right) + 8 = 0$$

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.

Find the value of

1. \(81 ^ { \frac { 1 } { 2 } }\),
2. \(81 ^ { \frac { 3 } { 4 } }\),
3. \(81 ^ { - \frac { 3 } { 4 } }\).

6. $$f ( x ) = \frac { ( 2 x + 1 ) ( x + 4 ) } { \sqrt { x } } , \quad x > 0$$

1. Show that \(\mathrm { f } ( x )\) can be written in the form \(P x ^ { \frac { 3 } { 2 } } + Q x ^ { \frac { 1 } { 2 } } + R x ^ { - \frac { 1 } { 2 } }\), stating the values of the constants \(P , Q\) and \(R\).
2. Find f \({ } ^ { \prime } ( x )\).
3. Show that the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) is parallel to the line with equation \(2 y = 11 x + 3\).
   (3)
   

   6. continued

   Leave blank

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { ( x + 2 ) ^ { \frac { 3 } { 2 } } } { 4 } , \quad x \geqslant - 2$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line with equation \(x = 10\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { ( x + 2 ) ^ { \frac { 3 } { 2 } } } { 4 }\)

1. Complete the table, giving values of \(y\) corresponding to \(x = 2\) and \(x = 6\)

   \(x\)	- 2	2	6	10
   \(y\)	0			\(6 \sqrt { } 3\)

2. Use the trapezium rule, with all the values of \(y\) from the completed table, to find an approximate value for the area of \(R\), giving your answer to 3 decimal places.

1. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of

$$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form. Given that $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { n } \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } = \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } }$$ (b) write down the value of \(n\).
(c) Hence, or otherwise, find the first 3 terms of the binomial expansion, in ascending powers of \(x\), of $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form.