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 and solve the given expressions, let’s break them down. a) i) To simplify \( \left(a-\frac{1}{a}\right)\left(a^{2}+1+\frac{1}{a^{2}}\right) \), we can expand it as follows: \[ = a(a^2 + 1 + \frac{1}{a^2}) - \frac{1}{a}(a^2 + 1 + \frac{1}{a^2}) \] \[ = a^3 + a - \frac{1}{a}(a^2 + 1) - 1 \] Combining terms gives us: \[ = a^3 + a - a - \frac{1}{a} - 1 \\ = a^3 - \frac{1}{a} - 1. \] ii) For \( \left(a-\frac{1}{a}\right)^{2} \): \[ = a^2 - 2 \cdot a \cdot \frac{1}{a} + \left(\frac{1}{a}\right)^{2} \\ = a^2 - 2 + \frac{1}{a^2} \\ = a^2 + \frac{1}{a^2} - 2. \] b) i) To calculate \( a^{2}+\frac{1}{a^{2}} \) given \( a-\frac{1}{a}=3 \), we know: \[ \left(a - \frac{1}{a}\right)^{2} = a^{2} - 2 + \frac{1}{a^{2}}. \] Thus, \[ 3^{2} = a^2 - 2 + \frac{1}{a^2} \implies 9 = a^{2} + \frac{1}{a^{2}} - 2 \implies a^{2} + \frac{1}{a^{2}} = 11. \] ii) To find \( a^{3}-\frac{1}{a^{3}} \), we use the identity: \[ a^{3} - \frac{1}{a^{3}} = \left(a - \frac{1}{a}\right)(a^{2} + 1 + \frac{1}{a^{2}}). \] With \( a - \frac{1}{a} = 3 \) and \( a^{2} + \frac{1}{a^{2}} = 11 \), we can calculate: \[ = 3 \left( 11 + 1 \right) = 3 \cdot 12 = 36. \] Thus, the final answers are: a) i) \( a^3 - \frac{1}{a} - 1 \), ii) \( a^2 + \frac{1}{a^2} - 2 \); b) i) \( 11 \), ii) \( 36 \).