EXERCISE 1.13 (a) Factorise: \( \begin{array}{lll}\text { (1) } 27 x^{3}-1 & \text { (2) } 8 x^{3}+1 & \text { (3) } 64 x^{3}-y^{3} \\ \text { (4) } 125-729 x^{3} & \text { (5) } a^{3} b^{3}-216 & \text { (6) } \\ \text { (65) } \\ \text { (7) } 8 a^{4}-64 a & \text { (8) }-x^{3}-27 & \text { (9) }\end{array} \)

Certainly! Let's factorise each of the given expressions step by step. We'll address each part from (1) to (8). It appears that part (6) and part (9) might be incomplete or missing, so we'll focus on the ones provided. --- ### **(1) \(27x^{3} - 1\)** **Type of Expression:** Difference of Cubes **Factorisation:** \[ 27x^{3} - 1 = (3x)^3 - 1^3 = (3x - 1)\left( (3x)^2 + 3x \cdot 1 + 1^2 \right) = (3x - 1)(9x^{2} + 3x + 1) \] --- ### **(2) \(8x^{3} + 1\)** **Type of Expression:** Sum of Cubes **Factorisation:** \[ 8x^{3} + 1 = (2x)^3 + 1^3 = (2x + 1)\left( (2x)^2 - 2x \cdot 1 + 1^2 \right) = (2x + 1)(4x^{2} - 2x + 1) \] --- ### **(3) \(64x^{3} - y^{3}\)** **Type of Expression:** Difference of Cubes **Factorisation:** \[ 64x^{3} - y^{3} = (4x)^3 - y^3 = (4x - y)\left( (4x)^2 + 4x \cdot y + y^2 \right) = (4x - y)(16x^{2} + 4xy + y^{2}) \] --- ### **(4) \(125 - 729x^{3}\)** **Type of Expression:** Difference of Cubes **Factorisation:** \[ 125 - 729x^{3} = 5^3 - (9x)^3 = (5 - 9x)\left(5^2 + 5 \cdot 9x + (9x)^2\right) = (5 - 9x)(25 + 45x + 81x^{2}) \] --- ### **(5) \(a^{3}b^{3} - 216\)** **Type of Expression:** Difference of Cubes **Factorisation:** \[ a^{3}b^{3} - 216 = (ab)^3 - 6^3 = (ab - 6)\left( (ab)^2 + ab \cdot 6 + 6^2 \right) = (ab - 6)(a^{2}b^{2} + 6ab + 36) \] --- ### **(7) \(8a^{4} - 64a\)** **Steps:** 1. **Factor out the Greatest Common Factor (GCF):** \[ 8a^{4} - 64a = 8a(a^{3} - 8) \] 2. **Recognise the Difference of Cubes:** \[ a^{3} - 8 = a^{3} - 2^{3} = (a - 2)(a^{2} + 2a + 4) \] 3. **Combine the Factors:** \[ 8a(a - 2)(a^{2} + 2a + 4) \] **Final Factorisation:** \[ 8a^{4} - 64a = 8a(a - 2)(a^{2} + 2a + 4) \] --- ### **(8) \(-x^{3} - 27\)** **Type of Expression:** Negative Sum of Cubes **Factorisation:** \[ -x^{3} - 27 = -\left( x^{3} + 3^{3} \right) = -\left( x + 3 \right)\left( x^{2} - 3x + 9 \right) = -(x + 3)(x^{2} - 3x + 9) \] --- ### **(6) & (9)** It seems that parts **(6)** and **(9)** of the exercise might be missing or incomplete. If you provide the complete expressions for these parts, I'd be happy to help factorise them as well! --- Feel free to reach out if you have any further questions or need clarification on any of the steps!