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]
we follow these steps:
1. First, simplify the division part involving [tex]\(\frac{2}{3} \div 2^4\)[/tex]:
[tex]\(2^4 = 16\)[/tex], so the expression becomes [tex]\(\frac{2}{3} \div 16\)[/tex].
To divide by 16, we multiply by its reciprocal, [tex]\(\frac{1}{16}\)[/tex]:
[tex]\[
\frac{2}{3} \div 16 = \frac{2}{3} \times \frac{1}{16} = \frac{2 \times 1}{3 \times 16} = \frac{2}{48} = \frac{1}{24}
\][/tex]
2. Next, address the addition [tex]\(\frac{3}{4} + \frac{1}{6}\)[/tex]:
To add these fractions, we need a common denominator. The least common multiple of 4 and 6 is 12.
Convert [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex] to fractions with a denominator of 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]
Now, add the two fractions:
[tex]\[
\frac{9}{12} + \frac{2}{12} = \frac{9 + 2}{12} = \frac{11}{12}
\][/tex]
3. Then, divide the result by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\left(\frac{11}{12}\right) \div \frac{1}{3} = \left(\frac{11}{12}\right) \times 3 = \frac{11}{12} \times \frac{3}{1} = \frac{11 \times 3}{12 \times 1} = \frac{33}{12} = \frac{11}{4}
\][/tex]
4. Finally, add the results of the two main parts:
[tex]\[
\frac{1}{24} + \frac{11}{4}
\][/tex]
Convert [tex]\(\frac{11}{4}\)[/tex] to a fraction with the same denominator as [tex]\(\frac{1}{24}\)[/tex]:
[tex]\[
\frac{11}{4} = \frac{11 \times 6}{4 \times 6} = \frac{66}{24}
\][/tex]
Now, add the two fractions:
[tex]\[
\frac{1}{24} + \frac{66}{24} = \frac{1 + 66}{24} = \frac{67}{24}
\][/tex]
So, the value of the expression is [tex]\(\boxed{\frac{67}{24}}\)[/tex].
