Simplify \(\left(\dfrac{12x^{-4}y^6}{27}\right)^{-\frac12}\) and write your answer in positive index notation.[2]
Given that \(2^{x+1}=2^x+32\), find \(x\).[2]
Write as a single fraction in its simplest form \(\dfrac{x+2}{x-3}-\dfrac{60}{2x^2-18}\).[4]
Explain why, for all integer values of \(n\), \(n(n^2-1)\) is always even.[2]
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Answer:(a) \(\dfrac{38a^4}{45b^4c^2}\) (b) \(\dfrac{3x^2}{2|y|^3}\); for \(y>0\), \(\dfrac{3x^2}{2y^3}\) (c) \(5\) (d) \(\dfrac{x+8}{x+3}\), \(x\ne\pm3\) (e) The product of three consecutive integers includes an even factor.