EXERCISE 7 Fully factorise each of the following. \( \begin{array}{ll}\text {-1. } 25 a^{2}-16 & \text { 2. } 9 b^{2}-1 \\ \text {-3. } 2 a^{2}+2 b^{2} & \text { 4. } 16 x^{2}-9 y^{2} \\ - \text { 5. } x^{2} y^{2}-1 & \text { 6. } 12 x^{8}-27 b^{6} \\ \text { 7. } 28 x^{3}-63 x b^{2} & \text { 8. } 125 m^{3}+15 m \\ \text { 9. }(2 x+3 y)^{2}-9 & \text { 10. }(3 a-2 b)^{2}-4 b^{2} \\ \text { 11. } 4 x^{2}-(3 x-2 y)^{2} & \text { 12. } 9 x^{2}(2 a-b)-16(2 a-b \\ \text { 13. } 25 a^{2}(a-3 b)+9(3 b-a) & \text { 14. } \frac{1}{3} x^{2}-3\end{array} \)

Factor the expression by following steps: - step0: Factor: \(2a^{2}+2b^{2}\) - step1: Factor the expression: \(2\left(a^{2}+b^{2}\right)\) Factor the expression \( 9 b^{2}-1 \). Factor the expression by following steps: - step0: Factor: \(9b^{2}-1\) - step1: Rewrite the expression: \(\left(3b\right)^{2}-1^{2}\) - step2: Factor the expression: \(\left(3b-1\right)\left(3b+1\right)\) Factor the expression \( 16 x^{2}-9 y^{2 \). Factor the expression by following steps: - step0: Factor: \(16x^{2}-9y^{2}\) - step1: Rewrite the expression: \(\left(4x\right)^{2}-\left(3y\right)^{2}\) - step2: Factor the expression: \(\left(4x-3y\right)\left(4x+3y\right)\) Factor the expression \( 12 x^{8}-27 b^{6 \). Factor the expression by following steps: - step0: Factor: \(12x^{8}-27b^{6}\) - step1: Factor the expression: \(3\left(4x^{8}-9b^{6}\right)\) - step2: Factor the expression: \(3\left(2x^{4}-3b^{3}\right)\left(2x^{4}+3b^{3}\right)\) Factor the expression \( x^{2} y^{2}-1 \). Factor the expression by following steps: - step0: Factor: \(x^{2}y^{2}-1\) - step1: Rewrite the expression: \(\left(xy\right)^{2}-1^{2}\) - step2: Factor the expression: \(\left(xy-1\right)\left(xy+1\right)\) Factor the expression \( (2 x+3 y)^{2}-9 \). Factor the expression by following steps: - step0: Factor: \(\left(2x+3y\right)^{2}-9\) - step1: Rewrite the expression: \(\left(2x+3y\right)^{2}-3^{2}\) - step2: Factor the expression: \(\left(2x+3y-3\right)\left(2x+3y+3\right)\) Factor the expression \( 125 m^{3}+15 m \). Factor the expression by following steps: - step0: Factor: \(125m^{3}+15m\) - step1: Rewrite the expression: \(5m\times 25m^{2}+5m\times 3\) - step2: Factor the expression: \(5m\left(25m^{2}+3\right)\) Factor the expression \( 28 x^{3}-63 x b^{2 \). Factor the expression by following steps: - step0: Factor: \(28x^{3}-63xb^{2}\) - step1: Factor the expression: \(7x\left(4x^{2}-9b^{2}\right)\) - step2: Factor the expression: \(7x\left(2x-3b\right)\left(2x+3b\right)\) Factor the expression \( \frac{1}{3} x^{2}-3 \). Factor the expression by following steps: - step0: Factor: \(\frac{1}{3}x^{2}-3\) - step1: Factor the expression: \(\frac{1}{3}\left(x^{2}-9\right)\) - step2: Factor the expression: \(\frac{1}{3}\left(x-3\right)\left(x+3\right)\) Factor the expression \( 4 x^{2}-(3 x-2 y)^{2 \). Factor the expression by following steps: - step0: Factor: \(4x^{2}-\left(3x-2y\right)^{2}\) - step1: Rewrite the expression: \(\left(2x\right)^{2}-\left(3x-2y\right)^{2}\) - step2: Factor the expression: \(\left(2x-\left(3x-2y\right)\right)\left(2x+3x-2y\right)\) - step3: Evaluate: \(\left(-x+2y\right)\left(2x+3x-2y\right)\) - step4: Evaluate: \(\left(-x+2y\right)\left(5x-2y\right)\) Factor the expression \( 9 x^{2}(2 a-b)-16(2 a-b \). Factor the expression by following steps: - step0: Factor: \(9x^{2}\left(2a-b\right)-16\left(2a-b\right)\) - step1: Factor the expression: \(\left(2a-b\right)\left(9x^{2}-16\right)\) - step2: Factor the expression: \(\left(2a-b\right)\left(3x-4\right)\left(3x+4\right)\) Factor the expression \( (3 a-2 b)^{2}-4 b^{2 \). Factor the expression by following steps: - step0: Factor: \(\left(3a-2b\right)^{2}-4b^{2}\) - step1: Rewrite the expression: \(\left(3a-2b\right)^{2}-\left(2b\right)^{2}\) - step2: Factor the expression: \(\left(3a-2b-2b\right)\left(3a-2b+2b\right)\) - step3: Evaluate: \(\left(3a-4b\right)\left(3a-2b+2b\right)\) - step4: Evaluate: \(\left(3a-4b\right)\times 3a\) Factor the expression \( 25 a^{2}-16 \). Factor the expression by following steps: - step0: Factor: \(25a^{2}-16\) - step1: Rewrite the expression: \(\left(5a\right)^{2}-4^{2}\) - step2: Factor the expression: \(\left(5a-4\right)\left(5a+4\right)\) Factor the expression \( 25 a^{2}(a-3 b)+9(3 b-a \). Factor the expression by following steps: - step0: Factor: \(25a^{2}\left(a-3b\right)+9\left(3b-a\right)\) - step1: Rewrite the expression: \(25a^{2}\left(-3b+a\right)-9\left(-3b+a\right)\) - step2: Factor the expression: \(\left(25a^{2}-9\right)\left(-3b+a\right)\) - step3: Factor the expression: \(\left(5a-3\right)\left(5a+3\right)\left(-3b+a\right)\) Here are the fully factored forms of each expression: 1. **\( 25 a^{2}-16 \)**: \[ (5a-4)(5a+4) \] 2. **\( 9 b^{2}-1 \)**: \[ (3b-1)(3b+1) \] 3. **\( 2 a^{2}+2 b^{2} \)**: \[ 2(a^{2}+b^{2}) \] 4. **\( 16 x^{2}-9 y^{2} \)**: \[ (4x-3y)(4x+3y) \] 5. **\( x^{2} y^{2}-1 \)**: \[ (xy-1)(xy+1) \] 6. **\( 12 x^{8}-27 b^{6} \)**: \[ 3(2x^{4}-3b^{3})(2x^{4}+3b^{3}) \] 7. **\( 28 x^{3}-63 x b^{2} \)**: \[ 7x(2x-3b)(2x+3b) \] 8. **\( 125 m^{3}+15 m \)**: \[ 5m(25m^{2}+3) \] 9. **\( (2 x+3 y)^{2}-9 \)**: \[ (2x+3y-3)(2x+3y+3) \] 10. **\( (3 a-2 b)^{2}-4 b^{2} \)**: \[ (3a-4b)(3a+4b) \] 11. **\( 4 x^{2}-(3 x-2 y)^{2} \)**: \[ (-x+2y)(5x-2y) \] 12. **\( 9 x^{2}(2 a-b)-16(2 a-b) \)**: \[ (2a-b)(3x-4)(3x+4) \] 13. **\( 25 a^{2}(a-3 b)+9(3 b-a) \)**: \[ (5a-3)(5a+3)(-3b+a) \] 14. **\( \frac{1}{3} x^{2}-3 \)**: \[ \frac{1}{3}(x-3)(x+3) \] These factorizations utilize various algebraic identities, including the difference of squares, common factors, and grouping techniques.