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 of the problem step by step. --- ## **4. a) Simplify:** ### **i) \( \left(a - \frac{1}{a}\right)\left(a^{2} + 1 + \frac{1}{a^{2}}\right) \)** **Solution:** Multiply the two expressions: \[ \left(a - \frac{1}{a}\right)\left(a^{2} + 1 + \frac{1}{a^{2}}\right) \] Distribute each term: \[ = a \cdot a^{2} + a \cdot 1 + a \cdot \frac{1}{a^{2}} - \frac{1}{a} \cdot a^{2} - \frac{1}{a} \cdot 1 - \frac{1}{a} \cdot \frac{1}{a^{2}} \] Simplify each term: \[ = a^{3} + a + \frac{1}{a} - a - \frac{1}{a} - \frac{1}{a^{3}} \] Combine like terms: \[ = a^{3} - \frac{1}{a^{3}} \] **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: \[ \left(a - \frac{1}{a}\right)^{2} = a^{2} - 2 \cdot a \cdot \frac{1}{a} + \left(\frac{1}{a}\right)^{2} \] Simplify each term: \[ = a^{2} - 2 + \frac{1}{a^{2}} \] **Answer:** \[ \left(a - \frac{1}{a}\right)^{2} = a^{2} + \frac{1}{a^{2}} - 2 \] --- ## **4. b) Calculate the value of:** ### **i) \( a^{2} + \frac{1}{a^{2}} \), if \( a - \frac{1}{a} = 3 \) ** **Given:** \[ a - \frac{1}{a} = 3 \] **Solution:** Square both sides of the equation to relate \( a^{2} + \frac{1}{a^{2}} \): \[ \left(a - \frac{1}{a}\right)^{2} = a^{2} - 2 \cdot a \cdot \frac{1}{a} + \frac{1}{a^{2}} = 9 \] Simplify: \[ a^{2} - 2 + \frac{1}{a^{2}} = 9 \] Add 2 to both sides: \[ a^{2} + \frac{1}{a^{2}} = 11 \] **Answer:** \[ a^{2} + \frac{1}{a^{2}} = 11 \] --- ### **ii) \( a^{3} - \frac{1}{a^{3}} \), if \( a - \frac{1}{a} = 3 \) ** **Given:** \[ a - \frac{1}{a} = 3 \quad \text{and} \quad a^{2} + \frac{1}{a^{2}} = 11 \] **Solution:** Use the identity for \( a^{3} - \frac{1}{a^{3}} \): \[ a^{3} - \frac{1}{a^{3}} = \left(a - \frac{1}{a}\right)\left(a^{2} + 1 + \frac{1}{a^{2}}\right) \] We already have \( a - \frac{1}{a} = 3 \) and \( a^{2} + \frac{1}{a^{2}} = 11 \), so: \[ a^{2} + 1 + \frac{1}{a^{2}} = 11 + 1 = 12 \] Multiply the terms: \[ a^{3} - \frac{1}{a^{3}} = 3 \times 12 = 36 \] **Alternative Method:** Alternatively, use the expansion of \( \left(a - \frac{1}{a}\right)^{3} \): \[ \left(a - \frac{1}{a}\right)^{3} = a^{3} - 3a + 3\frac{1}{a} - \frac{1}{a^{3}} \] This can be rearranged as: \[ a^{3} - \frac{1}{a^{3}} = \left(a - \frac{1}{a}\right)^{3} + 3\left(a - \frac{1}{a}\right) = 3^{3} + 3 \times 3 = 27 + 9 = 36 \] **Answer:** \[ a^{3} - \frac{1}{a^{3}} = 36 \] ---