Simplify the following expressions: (i) (27 3⁻²) (8 2⁻³) (ii) (76 * 7⁻³) / 74 (iii) 256^(-1/4) (225^(1/2) + 580)

Remember that any non-zero number raised to the power of 0 is 1.

$$3 \times 1$$

$$3$$

The simplified expression is $\boxed{3}$.

Problem (ii): $\frac{7^6 \times 7^{-3}}{7^4}$

$$\frac{1}{7}$$

The simplified expression is $\boxed{\frac{1}{7}}$.

Problem (iii): $256^{-\frac{1}{4}} (225^{\frac{1}{2}} + 58^0)$

For $256^{-\frac{1}{4}}$:

$$256^{-\frac{1}{4}} = (4^4)^{-\frac{1}{4}} = 4^{4 \times (-\frac{1}{4})} = 4^{-1} = \frac{1}{4}$$

For $225^{\frac{1}{2}}$:

$$225^{\frac{1}{2}} = \sqrt{225} = 15$$

For $58^0$:

$$58^0 = 1$$

$$\frac{16}{4} = 4$$

The simplified expression is $\boxed{4}$.

Problem (iv): $27^{\frac{2}{3}} \times 64^{\frac{1}{3}} \div 81^{\frac{1}{4}}$

For $27^{\frac{2}{3}}$:

$$27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^{3 \times \frac{2}{3}} = 3^2 = 9$$

For $64^{\frac{1}{3}}$:

$$64^{\frac{1}{3}} = (4^3)^{\frac{1}{3}} = 4^{3 \times \frac{1}{3}} = 4^1 = 4$$

For $81^{\frac{1}{4}}$:

$$81^{\frac{1}{4}} = (3^4)^{\frac{1}{4}} = 3^{4 \times \frac{1}{4}} = 3^1 = 3$$

$$12$$

The simplified expression is $\boxed{12}$.