1. **Statement of the problem:** Calculate various results for the given sets of numbers and expressions. 2. Calculate the number of elements $N$ in the sequence $420, 380, 388, 400, 420$. $$N = 5$$ 3. Calculate $B = 2\sqrt{175} - 7\sqrt{63} + 112\sqrt{112}$. Simplify each term: $\sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7}$ $\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}$ $\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}$ Then: $B = 2 \times 5\sqrt{7} - 7 \times 3\sqrt{7} + 112 \times 4 \sqrt{7}$ $B = 10\sqrt{7} - 21\sqrt{7} + 448\sqrt{7} = (10 - 21 + 448)\sqrt{7} = 437\sqrt{7}$ 4. Calculate $A = 2\Theta(2^{40} \times 1.5) + x(2^{-38})^3$. Assuming $\Theta$ is a function placeholder and $x$ is a variable, evaluate powers. First compute $(2^{-38})^3 = 2^{-114}$. As we don't have the exact meaning of $\Theta$ or $x$, leave $A$ as $A = 2\Theta(2^{40} \times 1.5) + x \times 2^{-114}$. 5. Calculate $C = \sqrt{3} + \frac{1}{2}\sqrt{3} = \frac{3}{2} \sqrt{3}$. 6. Given $E = 1.783 \times 10^8$. 7. Calculate real number for $197$ as follows: $A = 2.1326 \times 10^2 = 213.26$ $H = \sqrt{217} + 20^2 = \sqrt{217} + 400 \approx 14.73 + 400 = 414.73$ $G = (0.011 \times (2.17)^2) \times (0.012 \times (0.175)^3)$ Calculate each part: $2.17^2 = 4.7089$ $0.175^3 = 0.005359$ Then: $G = (0.011 \times 4.7089) \times (0.012 \times 0.005359) = 0.051798 \times 0.0000643 \approx 3.33 \times 10^{-6}$ 8. Calculate $D = (\frac{1}{10})^{-1} \times \frac{9}{17} = 10 \times \frac{9}{17} = \frac{90}{17} \approx 5.29$. 9. Given $B = \frac{1}{2} \times C^{-\frac{1}{7}}$, substitute $C = \frac{3}{2} \sqrt{3} \approx 2.598$. Calculate $C^{-\frac{1}{7}} = (2.598)^{-0.142857} \approx 0.889$. So, $B = \frac{1}{2} \times 0.889 = 0.4445$ 10. Compute $E = \sqrt{2} - \sqrt{2} - 1 = 0 - 1 = -1$. **Final results:** $N=5$ $B = 437\sqrt{7} \approx 1155.46$ $A = 2\Theta(2^{40} \times 1.5) + x \times 2^{-114}$ (cannot simplify without $\Theta$ and $x$) $C = \frac{3}{2} \sqrt{3} \approx 2.598$ $E = 1.783 \times 10^8$ $A = 213.26$ $H \approx 414.73$ $G \approx 3.33 \times 10^{-6}$ $D \approx 5.29$ $B \approx 0.4445$ (redefined in step 9) $E = -1$