Задание 4. Найдите значение выражения: 1) sqrt(a^2 + 8ab + 16b^2) при a=3 3/7, b=1/7...

Для решения этих примеров мы будем использовать формулы квадрата суммы $(x+y)^2 = x^2+2xy+y^2$ и квадрата разности $(x-y)^2 = x^2-2xy+y^2$, а также свойство корня $\sqrt{x^2} = |x|$. 1. $\sqrt{a^2+8ab+16b^2} = \sqrt{(a+4b)^2} = |a+4b|$ При $a=3\frac{3}{7}, b=\frac{1}{7}$: $|3\frac{3}{7} + 4 \cdot \frac{1}{7}| = |3\frac{3}{7} + \frac{4}{7}| = |3\frac{7}{7}| = 4$ 2. $\sqrt{a^2+12ab+36b^2} = \sqrt{(a+6b)^2} = |a+6b|$ При $a=7\frac{2}{5}, b=\frac{3}{5}$: $|7\frac{2}{5} + 6 \cdot \frac{3}{5}| = |7\frac{2}{5} + \frac{18}{5}| = |7\frac{2}{5} + 3\frac{3}{5}| = |10\frac{5}{5}| = 11$ 3. $\sqrt{a^2+10ab+25b^2} = \sqrt{(a+5b)^2} = |a+5b|$ При $a=1\frac{6}{13}, b=\frac{4}{13}$: $|1\frac{6}{13} + 5 \cdot \frac{4}{13}| = |1\frac{6}{13} + \frac{20}{13}| = |1\frac{6}{13} + 1\frac{7}{13}| = |2\frac{13}{13}| = 3$ 4. $\sqrt{a^2+8ab+16b^2} = \sqrt{(a+4b)^2} = |a+4b|$ При $a=3\frac{2}{3}, b=\frac{1}{3}$: $|3\frac{2}{3} + 4 \cdot \frac{1}{3}| = |3\frac{2}{3} + 1\frac{1}{3}| = |4\frac{3}{3}| = 5$ 5. $\sqrt{9a^2+6ab+b^2} = \sqrt{(3a+b)^2} = |3a+b|$ При $a=\frac{5}{13}, b=6\frac{11}{13}$: $|3 \cdot \frac{5}{13} + 6\frac{11}{13}| = |\frac{15}{13} + 6\frac{11}{13}| = |1\frac{2}{13} + 6\frac{11}{13}| = 8$ 6. $\sqrt{16a^2+8ab+b^2} = \sqrt{(4a+b)^2} = |4a+b|$ При $a=\frac{3}{11}, b=5\frac{10}{11}$: $|4 \cdot \frac{3}{11} + 5\frac{10}{11}| = |\frac{12}{11} + 5\frac{10}{11}| = |1\frac{1}{11} + 5\frac{10}{11}| = 7$ 7. $\sqrt{25a^2+10ab+b^2} = \sqrt{(5a+b)^2} = |5a+b|$ При $a=\frac{4}{9}, b=3\frac{7}{9}$: $|5 \cdot \frac{4}{9} + 3\frac{7}{9}| = |\frac{20}{9} + 3\frac{7}{9}| = |2\frac{2}{9} + 3\frac{7}{9}| = 6$ 8. $\sqrt{36a^2+12ab+b^2} = \sqrt{(6a+b)^2} = |6a+b|$ При $a=\frac{4}{5}, b=8\frac{1}{5}$: $|6 \cdot \frac{4}{5} + 8\frac{1}{5}| = |\frac{24}{5} + 8\frac{1}{5}| = |4\frac{4}{5} + 8\frac{1}{5}| = 13$ 9. $\sqrt{a^2-6ab+9b^2} = \sqrt{(a-3b)^2} = |a-3b|$ При $a=3, b=6$: $|3 - 3 \cdot 6| = |3 - 18| = |-15| = 15$ 10. $\sqrt{a^2-12ab+36b^2} = \sqrt{(a-6b)^2} = |a-6b|$ При $a=8, b=3$: $|8 - 6 \cdot 3| = |8 - 18| = |-10| = 10$ 11. $\sqrt{a^2-8ab+16b^2} = \sqrt{(a-4b)^2} = |a-4b|$ При $a=4, b=3$: $|4 - 4 \cdot 3| = |4 - 12| = |-8| = 8$ 12. $\sqrt{a^2-10ab+25b^2} = \sqrt{(a-5b)^2} = |a-5b|$ При $a=7, b=2$: $|7 - 5 \cdot 2| = |7 - 10| = |-3| = 3$ 13. $\sqrt{a^2+10ab+25b^2} = \sqrt{(a+5b)^2} = |a+5b|$ При $a=8, b=-2$: $|8 + 5 \cdot (-2)| = |8 - 10| = |-2| = 2$ 14. $\sqrt{a^2+6ab+9b^2} = \sqrt{(a+3b)^2} = |a+3b|$ При $a=5, b=-4$: $|5 + 3 \cdot (-4)| = |5 - 12| = |-7| = 7$ 15. $\sqrt{a^2+12ab+36b^2} = \sqrt{(a+6b)^2} = |a+6b|$ При $a=7, b=-3$: $|7 + 6 \cdot (-3)| = |7 - 18| = |-11| = 11$ 16. $\sqrt{a^2+4ab+4b^2} = \sqrt{(a+2b)^2} = |a+2b|$ При $a=2, b=-4$: $|2 + 2 \cdot (-4)| = |2 - 8| = |-6| = 6$