The value of $$\frac{3 \div \left\{5 - 5 \div (6 - 7) \times 8 + 9\right\}}{4 + 4 \times 4 \div 4  of  4}$$ is:

Apply BODMAS rule to this type of problems

According to Bodmas rule, if an expression contains brackets ((), {}, []) we have to first solve or simplify the bracket followed by of (powers and roots etc.), then division, multiplication, addition and subtraction from left to right.

Numerator = $$3 \div \left\{5-5\div(6-7)\times8+9\right\} = 3\div \left\{5-5\div(-1)\times8+9\right\}$$
$$= 3\div\left\{5+5\times8+9\right\} = 3\div\left\{5+40+9\right\}$$
$$=\dfrac{3}{54} = \dfrac{1}{18}$$

Denominator = $$4+4\times4\div4 \text{of} 4 = 4+4\times4\div16 = 4+1 = 5$$

$$\dfrac{3 \div \left\{5 - 5 \div (6 - 7) \times 8 + 9\right\}}{4 + 4 \times 4 \div 4 of 4} = \dfrac{(\dfrac{1}{18})}{5} = \dfrac{1}{90}$$