- Simplify fully the following expressions:
i. \(\frac { 7 y ^ { 13 } } { 35 y ^ { 7 } }\)
ii. \(6 x ^ { - 2 } \div x ^ { - 5 }\) - A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) is defined by \(u _ { 1 } = 7\) and \(u _ { n + 1 } = u _ { n } + 4\) for \(n \geq 1\).
i. State what type of sequence this is.
ii. Find \(u _ { 17 }\). - i. Write \(3 x ^ { 2 } - 6 x + 1\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are integers.
ii. Solve \(3 x ^ { 2 } - 6 x + 1 \leq 0\), giving your answer in set notation. In this question you must show detailed reasoning.
i. Express \(\frac { \sqrt { 2 } } { 1 - \sqrt { 2 } }\) in the form \(c + d \sqrt { } e\), where \(c\) and \(d\) are integers and \(e\) is a prime number.
ii. Solve the equation \(\left( 8 p ^ { 6 } \right) ^ { \frac { 1 } { 3 } } = 8\). - Let \(a = \log _ { 2 } x , b = \log _ { 2 } y\) and \(c = \log _ { 2 } z\).
Express \(\log _ { 2 } ( x y ) - \log _ { 2 } \left( \frac { z } { x ^ { 2 } } \right)\) in terms of \(a , b\) and \(c\).