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) \( a^{3}-\frac{1}{a^{3}} \).

Let's solve each part step by step. --- ### **a) Simplify:** #### **i) \( \left(a - \frac{1}{a}\right)\left(a^{2} + 1 + \frac{1}{a^{2}}\right) \)** **Solution:** Multiply the two expressions: \[ \begin{align*} \left(a - \frac{1}{a}\right)\left(a^{2} + 1 + \frac{1}{a^{2}}\right) &= a \cdot a^{2} + a \cdot 1 + a \cdot \frac{1}{a^{2}} - \frac{1}{a} \cdot a^{2} - \frac{1}{a} \cdot 1 - \frac{1}{a} \cdot \frac{1}{a^{2}} \\ &= a^{3} + a + \frac{1}{a} - a - \frac{1}{a} - \frac{1}{a^{3}} \\ &= a^{3} - \frac{1}{a^{3}} \end{align*} \] **Answer:** \[ \left(a - \frac{1}{a}\right)\left(a^{2} + 1 + \frac{1}{a^{2}}\right) = a^{3} - \frac{1}{a^{3}} \] --- #### **ii) \( \left(a - \frac{1}{a}\right)^{2} \)** **Solution:** Expand the square: \[ \begin{align*} \left(a - \frac{1}{a}\right)^{2} &= a^{2} - 2 \cdot a \cdot \frac{1}{a} + \left(\frac{1}{a}\right)^{2} \\ &= a^{2} - 2 + \frac{1}{a^{2}} \end{align*} \] **Answer:** \[ \left(a - \frac{1}{a}\right)^{2} = a^{2} - 2 + \frac{1}{a^{2}} \] --- ### **b) Calculate the value of:** #### **i) \( a^{2} + \frac{1}{a^{2}} \), if \( a - \frac{1}{a} = 3 \)** **Solution:** Start by squaring both sides of the given equation: \[ \begin{align*} \left(a - \frac{1}{a}\right)^{2} &= 3^{2} \\ a^{2} - 2 + \frac{1}{a^{2}} &= 9 \quad \text{(Using the expansion from part a.ii)} \\ a^{2} + \frac{1}{a^{2}} &= 9 + 2 \\ a^{2} + \frac{1}{a^{2}} &= 11 \end{align*} \] **Answer:** \[ a^{2} + \frac{1}{a^{2}} = 11 \] --- #### **ii) \( a^{3} - \frac{1}{a^{3}} \)** **Solution:** Use the identity for the cube of a binomial: \[ \left(a - \frac{1}{a}\right)^{3} = a^{3} - 3a + 3\cdot\frac{1}{a} - \frac{1}{a^{3}} \] Rearrange to solve for \( a^{3} - \frac{1}{a^{3}} \): \[ \begin{align*} \left(a - \frac{1}{a}\right)^{3} &= a^{3} - \frac{1}{a^{3}} - 3\left(a - \frac{1}{a}\right) \\ a^{3} - \frac{1}{a^{3}} &= \left(a - \frac{1}{a}\right)^{3} + 3\left(a - \frac{1}{a}\right) \end{align*} \] Given that \( a - \frac{1}{a} = 3 \): \[ \begin{align*} a^{3} - \frac{1}{a^{3}} &= 3^{3} + 3 \times 3 \\ &= 27 + 9 \\ &= 36 \end{align*} \] **Answer:** \[ a^{3} - \frac{1}{a^{3}} = 36 \] --- ### **Final Answers:** - **a)** i) \( a^{3} - \dfrac{1}{a^{3}} \) ii) \( a^{2} - 2 + \dfrac{1}{a^{2}} \) - **b)** i) \( a^{2} + \dfrac{1}{a^{2}} = 11 \) ii) \( a^{3} - \dfrac{1}{a^{3}} = 36 \)