15. \( \frac{1}{2} a^{2}-8 b^{6} \) 16. \( 4(2 a-1)^{2}-36(a-3)^{2} \) 17. \( 2 x^{8}-2 y^{4} \) 18. \( \frac{1}{5} x^{3} y-20 x y^{3} \) 19. \( 9 m^{2}(2 p+3)-4(2 p+3) \) -20. \( 3 k\left(k^{2}-m^{2}\right)-3 k(m-k) \) You can use factorising to make it easier to simplify. For example, if you ar allowed to use a calculator it is easier to simplify \( 39^{2}-29^{2} \) if you factorise

Factor the expression by following steps: - step0: Factor: \(\frac{1}{2}a^{2}-8b^{6}\) - step1: Factor the expression: \(\frac{1}{2}\left(a^{2}-16b^{6}\right)\) - step2: Factor the expression: \(\frac{1}{2}\left(a-4b^{3}\right)\left(a+4b^{3}\right)\) Factor the expression \( 2 x^{8}-2 y^{4} \). Factor the expression by following steps: - step0: Factor: \(2x^{8}-2y^{4}\) - step1: Factor the expression: \(2\left(x^{8}-y^{4}\right)\) - step2: Factor the expression: \(2\left(x^{4}-y^{2}\right)\left(x^{4}+y^{2}\right)\) - step3: Evaluate: \(2\left(x^{2}-y\right)\left(x^{2}+y\right)\left(x^{4}+y^{2}\right)\) Factor the expression \( \frac{1}{5} x^{3} y-20 x y^{3} \). Factor the expression by following steps: - step0: Factor: \(\frac{1}{5}x^{3}y-20xy^{3}\) - step1: Factor the expression: \(\frac{1}{5}xy\left(x^{2}-100y^{2}\right)\) - step2: Factor the expression: \(\frac{1}{5}xy\left(x-10y\right)\left(x+10y\right)\) Factor the expression \( 3 k\left(k^{2}-m^{2}\right)-3 k(m-k) \). Factor the expression by following steps: - step0: Factor: \(3k\left(k^{2}-m^{2}\right)-3k\left(m-k\right)\) - step1: Rewrite the expression: \(3k\left(k-m\right)\left(k+m\right)+3k\left(k-m\right)\) - step2: Factor the expression: \(3k\left(k-m\right)\left(k+m+1\right)\) Factor the expression \( 4(2 a-1)^{2}-36(a-3)^{2} \). Factor the expression by following steps: - step0: Factor: \(4\left(2a-1\right)^{2}-36\left(a-3\right)^{2}\) - step1: Factor the expression: \(4\left(\left(2a-1\right)^{2}-9\left(a-3\right)^{2}\right)\) - step2: Factor the expression: \(4\left(\left(2a-1\right)+3\left(a-3\right)\right)\left(\left(2a-1\right)-3\left(a-3\right)\right)\) - step3: Calculate: \(4\left(5a-10\right)\left(-a+8\right)\) - step4: Factor the expression: \(4\times 5\left(a-2\right)\left(-a+8\right)\) - step5: Simplify the expression: \(20\left(a-2\right)\left(-a+8\right)\) Factor the expression \( 9 m^{2}(2 p+3)-4(2 p+3) \). Factor the expression by following steps: - step0: Factor: \(9m^{2}\left(2p+3\right)-4\left(2p+3\right)\) - step1: Factor the expression: \(\left(2p+3\right)\left(9m^{2}-4\right)\) - step2: Factor the expression: \(\left(2p+3\right)\left(3m-2\right)\left(3m+2\right)\) Here are the factorizations for each expression: 15. For the expression \( \frac{1}{2} a^{2}-8 b^{6} \): \[ \frac{1}{2}(a-4b^{3})(a+4b^{3}) \] 16. For the expression \( 4(2 a-1)^{2}-36(a-3)^{2} \): \[ 20(a-2)(-a+8) \] 17. For the expression \( 2 x^{8}-2 y^{4} \): \[ 2(x^{2}-y)(x^{2}+y)(x^{4}+y^{2}) \] 18. For the expression \( \frac{1}{5} x^{3} y-20 x y^{3} \): \[ \frac{1}{5}xy(x-10y)(x+10y) \] 19. For the expression \( 9 m^{2}(2 p+3)-4(2 p+3) \): \[ (2p+3)(3m-2)(3m+2) \] 20. For the expression \( 3 k\left(k^{2}-m^{2}\right)-3 k(m-k) \): \[ 3k(k-m)(k+m+1) \] These factorizations simplify the expressions and make further calculations easier.