To simplify the given expression [tex]\(\frac{2}{3} \div 2^4 + \left( \frac{3}{4} + \frac{1}{6} \right) \div \frac{1}{3}\)[/tex], follow these steps:
1. Simplify [tex]\(\frac{2}{3} \div 2^4\)[/tex]:
[tex]\[
\frac{2}{3} \div 2^4 = \frac{2}{3} \div 16 = \frac{2}{3} \times \frac{1}{16} = \frac{2 \times 1}{3 \times 16} = \frac{2}{48} = \frac{1}{24}
\][/tex]
2. Simplify [tex]\(\frac{3}{4} + \frac{1}{6}\)[/tex]:
[tex]\[
\frac{3}{4} + \frac{1}{6} = \frac{3 \times 3 + 1 \times 2}{4 \times 3} = \frac{9 + 2}{12} = \frac{11}{12}
\][/tex]
3. Divide [tex]\(\frac{11}{12}\)[/tex] by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{11}{12} \div \frac{1}{3} = \frac{11}{12} \times \frac{3}{1} = \frac{11 \times 3}{12 \times 1} = \frac{33}{12} = \frac{33}{12} = \frac{33 \div 3}{12 \div 3} = \frac{11}{4}
\][/tex]
4. Combine the results:
[tex]\[
\frac{1}{24} + \frac{11}{4}
\][/tex]
5. Simplify [tex]\(\frac{1}{24} + \frac{11}{4}\)[/tex]:
First, find a common denominator:
[tex]\[
\frac{1}{24} + \frac{11 \times 6}{4 \times 6} = \frac{1}{24} + \frac{66}{24} = \frac{1 + 66}{24} = \frac{67}{24}
\][/tex]
Therefore, the expression evaluates to [tex]\(\frac{67}{24}\)[/tex].
