1. The problem asks to evaluate the sum $a = \sqrt{\frac{19}{2,(1)}} + \sqrt{\frac{20}{2,(2)}} + \sqrt{\frac{21}{2,(3)}} + \dots + \sqrt{\frac{26}{2,(8)}}$. 2. We need to clarify the notation $2,(k)$. Assuming this means the denominator is $2$ multiplied by the number in parentheses, so the denominator for each term is $2 \times k$ where $k$ runs from 1 to 8. 3. Therefore, the expression becomes: $$a = \sqrt{\frac{19}{2 \times 1}} + \sqrt{\frac{20}{2 \times 2}} + \sqrt{\frac{21}{2 \times 3}} + \sqrt{\frac{22}{2 \times 4}} + \sqrt{\frac{23}{2 \times 5}} + \sqrt{\frac{24}{2 \times 6}} + \sqrt{\frac{25}{2 \times 7}} + \sqrt{\frac{26}{2 \times 8}}$$ 4. Simplify each term inside the square roots: $\sqrt{\frac{19}{2}} + \sqrt{\frac{20}{4}} + \sqrt{\frac{21}{6}} + \sqrt{\frac{22}{8}} + \sqrt{\frac{23}{10}} + \sqrt{\frac{24}{12}} + \sqrt{\frac{25}{14}} + \sqrt{\frac{26}{16}}$ 5. Calculate each term: - $\sqrt{\frac{19}{2}} = \sqrt{9.5} \approx 3.082$ - $\sqrt{\frac{20}{4}} = \sqrt{5} \approx 2.236$ - $\sqrt{\frac{21}{6}} = \sqrt{3.5} \approx 1.871$ - $\sqrt{\frac{22}{8}} = \sqrt{2.75} \approx 1.658$ - $\sqrt{\frac{23}{10}} = \sqrt{2.3} \approx 1.517$ - $\sqrt{\frac{24}{12}} = \sqrt{2} \approx 1.414$ - $\sqrt{\frac{25}{14}} \approx \sqrt{1.7857} \approx 1.336$ - $\sqrt{\frac{26}{16}} = \sqrt{1.625} \approx 1.275$ 6. Sum all terms: $$a \approx 3.082 + 2.236 + 1.871 + 1.658 + 1.517 + 1.414 + 1.336 + 1.275 = 14.389$$ Final answer: $$a \approx 14.39$$