The value of \(\frac{1}{2^2}+\frac{1}{2^3} \div\left(\frac{1}{2}+\frac{1}{2} \div 1\right) \) + \(3 \frac{1}{3} \div \frac{5}{2} \times \frac{3}{4} \div 6 \frac{2}{3} \times \frac{7}{6}\) is:

Given:

The value of \(\dfrac{1}{2^2}+\dfrac{1}{2^3} \div \left(\dfrac{1}{2}+\dfrac{1}{2} \div 1\right) + 3 \frac{1}{3} \div \frac{5}{2} \times \frac{3}{4} \div 6 \frac{2}{3} \times \frac{7}{6}\)

Formula used:

Apply basic arithmetic operations and simplification rules.

Calculation:

\(\dfrac{1}{2^2}+\dfrac{1}{2^3} \div \left(\dfrac{1}{2}+\dfrac{1}{2} \div 1\right) + 3 \frac{1}{3} \div \frac{5}{2} \times \frac{3}{4} \div 6 \frac{2}{3} \times \frac{7}{6}\)

First, simplify each part separately:

\(\dfrac{1}{2^2} = \dfrac{1}{4}\)

\(\dfrac{1}{2^3} = \dfrac{1}{8}\)

\(\dfrac{1}{2} + \dfrac{1}{2} \div 1 = \dfrac{1}{2} + \dfrac{1}{2} = 1\)

\(3 \frac{1}{3} \div \frac{5}{2} = \frac{10}{3} \times \frac{2}{5} = \frac{4}{3}\)

\(\frac{3}{4} \div 6 \frac{2}{3} = \frac{3}{4} \times \frac{3}{20} = \frac{9}{80}\)

\(\frac{4}{3} \times \frac{9}{80} \times \frac{7}{6} = \frac{7}{40}\)

Combining the parts:

\(\left(\frac{1}{4} + \frac{1}{8} \div 1 + \frac{7}{40}\right)\)

\(\left(\frac{1}{4} + \frac{1}{8} + \frac{7}{40}\right)\)

Find a common denominator (LCM of 4, 8, 40):

\((\frac{10+5+7}{40})\)

⇒ 22 / 40

⇒ 11 / 20 

∴ The correct answer is option (3).