Simplify and evaluate the following expressions: 1) 2(y-3)(2-y) 2) (4/(81))3 3) 2ab - 3b + 10a - 15 4) (1)/(2)x2 − 8 = 0

$$(y-3)(2-y) = y(2) + y(-y) -3(2) -3(-y)$$ $$= 2y - y^2 - 6 + 3y$$ $$= -y^2 + 5y - 6$$

$$2(-y^2 + 5y - 6) = 2(-y^2) + 2(5y) + 2(-6)$$ $$= -2y^2 + 10y - 12$$

The simplified expression is:

$$\boxed{-2y^2 + 10y - 12}$$

2. Evaluate $\left(\sqrt[4]{\frac{16}{81}}\right)^3$

$$\sqrt[4]{\frac{16}{81}} = \frac{\sqrt[4]{16}}{\sqrt[4]{81}}$$

Since $2^4 = 16$ and $3^4 = 81$:

$$= \frac{2}{3}$$

$$\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3}$$ $$= \frac{8}{27}$$

The evaluated expression is:

$$\boxed{\frac{8}{27}}$$

3. Factorise Completely $2ab - 3b + 10a - 15$

From the first group, factor out $b$:

$$b(2a - 3)$$

From the second group, factor out 5:

$$5(2a - 3)$$

$$b(2a - 3) + 5(2a - 3) = (2a - 3)(b + 5)$$

The completely factorised expression is:

$$\boxed{(2a - 3)(b + 5)}$$

4. Solve the equation $\frac{1}{2}x^2 - 8 = 0$

$$x^2 = 8 \times 2$$ $$x^2 = 16$$

Remember to include both positive and negative roots.

$$x = \pm\sqrt{16}$$ $$x = \pm 4$$

The solutions for $x$ are:

$$\boxed{x = 4 \text{ or } x = -4}$$