c. \( 2,6+\left(\frac{1}{3}+1\right)^{2}-\sqrt[3]{\frac{8}{27}}: \sqrt[4]{\frac{81}{16}}= \)

First, let's break down the expression step by step. We start with \( \frac{1}{3}+1 = \frac{1}{3} + \frac{3}{3} = \frac{4}{3} \). Squaring this gives us \( \left(\frac{4}{3}\right)^{2} = \frac{16}{9} \). Next, we simplify the cube root: \( \sqrt[3]{\frac{8}{27}} = \frac{2}{3} \). Now we look at the fourth root: \( \sqrt[4]{\frac{81}{16}} = \sqrt[4]{\frac{3^4}{4^2}} = \frac{3}{2}. \) Putting it all together, we have: \[ c = 2.6 + \frac{16}{9} - \frac{2}{3} : \frac{3}{2} \] We first convert \( 2.6 \) into a fraction: \( 2.6 = \frac{26}{10} = \frac{13}{5} \). Now let’s find a common denominator for the fractions involved. The common denominator of 9, 3, and 2 gives us 18: Now we compute: 1. \( \frac{2}{3} = \frac{12}{18} \) 2. \( \frac{16}{9} = \frac{32}{18} \) 3. \( \frac{3}{2} = \frac{27}{18} \) Combining all these, we get: \[ c = \frac{13}{5} + \frac{32}{18} - \frac{12}{18} \div \frac{27}{18} \] Now, simplifying the division part: \[ \frac{12}{18} \div \frac{27}{18} = \frac{12}{18} \times \frac{18}{27} = \frac{12}{27} = \frac{4}{9} \] Finally, putting it all together, we need to handle \( c = \frac{13}{5} + \frac{32}{18} - \frac{4}{9} \). After a suitable common denominator (which is 90), we compute: \[ \frac{13}{5} = \frac{234}{90}, \quad \frac{32}{18} = \frac{160}{90}, \quad \frac{4}{9} = \frac{40}{90} \] Final calculation: \[ c = \frac{234 + 160 - 40}{90} = \frac{354}{90} \] Reducing gives: \[ c = \frac{59}{15} \] So the final answer is: \[ c = \frac{59}{15} \] or approximately \(3.9333\).