To simplify the expression [tex]\(\frac{2}{3} \div 2^4+\left(\frac{3}{4}+\frac{1}{6}\right) \div \frac{1}{3}\)[/tex], let's go through its simplification step by step.
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]\(\left(\frac{3}{4}+\frac{1}{6}\right)\)[/tex]:
- To add these fractions, find a common denominator. The least common multiple of 4 and 6 is 12.
[tex]\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
\][/tex]
[tex]\[
\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}
\][/tex]
- Add these fractions:
[tex]\[
\frac{9}{12} + \frac{2}{12} = \frac{11}{12}
\][/tex]
3. Divide the result of step 2 by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\left(\frac{11}{12}\right) \div \frac{1}{3} = \left(\frac{11}{12}\right) \times 3 = \frac{11 \times 3}{12} = \frac{33}{12} = \frac{11}{4} = 2\frac{3}{4} \approx 2.75
\][/tex]
4. Add the results of step 1 and step 3:
[tex]\[
\frac{1}{24} + \frac{11}{4}
\][/tex]
- Find a common denominator for the fractions. The least common multiple of 24 and 4 is 24.
[tex]\[
\frac{11}{4} = \frac{11 \times 6}{4 \times 6} = \frac{66}{24}
\][/tex]
- Add these fractions:
[tex]\[
\frac{1}{24} + \frac{66}{24} = \frac{1 + 66}{24} = \frac{67}{24}
\][/tex]
Therefore, the simplified value of the expression is:
[tex]\[
\boxed{\frac{67}{24}}
\][/tex]
