4. Complete the following. a) Simplify: i) \( \left(a-\frac{1}{a}\right)\left(a^{2}+1+\frac{1}{a^{2}}\right) \) ii) \( \left(a-\frac{1}{a}\right)^{2} \). b) Calculate the value of: i) \( a^{2}+\frac{1}{a^{2}} \), if \( a-\frac{1}{a}=3 \) ii) \( a^{3}-\frac{1}{a^{3}} \)

To simplify \( \left(a-\frac{1}{a}\right)\left(a^{2}+1+\frac{1}{a^{2}}\right) \), use the fact that \( a^2 + 1 + \frac{1}{a^2} = (a-\frac{1}{a})^2 + 2 \). Expanding gives \((a - \frac{1}{a})((a - \frac{1}{a})^2 + 2)\). Substituting \( x = a - \frac{1}{a} \), we can write it as \( x(x^2 + 2) = x^3 + 2x\). For part ii, \( \left(a-\frac{1}{a}\right)^{2} \) simplifies to \( a^2 - 2 + \frac{1}{a^2} = (a^2 + \frac{1}{a^2}) - 2 \). If \( a - \frac{1}{a} = 3 \), we first find \( a^2 + \frac{1}{a^2} \) using \( (a - \frac{1}{a})^2 = a^2 - 2 + \frac{1}{a^2} \rightarrow 9 = (a^2 + \frac{1}{a^2}) - 2 \). Thus, \( a^2 + \frac{1}{a^2} = 11 \). To find \( a^3 - \frac{1}{a^3} \), use the identity \( a^3 - \frac{1}{a^3} = (a - \frac{1}{a)((a^2 + \frac{1}{a^2})} \). Substitute \( a - \frac{1}{a} = 3 \) and \( a^2 + \frac{1}{a^2} = 11\). Thus, \( a^3 - \frac{1}{a^3} = 3 \times 11 = 33 \).