Номер 31.5, страница 184 - гдз по алгебре 7 класс учебник Абылкасымова, Кучер

31.5. Вычислите с помощью формулы $a^2 - b^2 = (a - b)(a + b)$:

1) $13^2 - 9^2$;

2) $20^2 - 19^2$;

3) $2.2^2 - 2.8^2$;

4) $3.5^2 - 3.7^2$;

5) $\left(\frac{5}{6}\right)^2 - \left(\frac{2}{3}\right)^2$;

6) $\left(\frac{7}{9}\right)^2 - \left(\frac{1}{6}\right)^2$;

7) $\left(\frac{5}{12}\right)^2 - \left(\frac{3}{4}\right)^2$;

8) $\left(\frac{3}{10}\right)^2 - \left(\frac{4}{5}\right)^2$;

9) $\left(\frac{8}{15}\right)^2 - \left(\frac{4}{5}\right)^2$;

10) $\left(2\frac{1}{7}\right)^2 - \left(2\frac{1}{7}\right)^2$;

11) $\left(3\frac{1}{3}\right)^2 - \left(4\frac{1}{2}\right)^2$;

12) $\left(5\frac{1}{6}\right)^2 - \left(7\frac{1}{3}\right)^2$;

13) $51^2 - 41^2$;

14) $54^2 - 46^2$;

15) $76^2 - 24^2$;

16) $328^2 - 172^2$;

17) $\left(3\frac{2}{3}\right)^2 - \left(2\frac{1}{3}\right)^2$;

18) $\left(7\frac{5}{9}\right)^2 - \left(4\frac{4}{9}\right)^2$.

1) $13^2 - 9^2 = (13 - 9)(13 + 9) = 4 \cdot 22 = 88$.

Ответ: 88

2) $20^2 - 19^2 = (20 - 19)(20 + 19) = 1 \cdot 39 = 39$.

Ответ: 39

3) $2,2^2 - 2,8^2 = (2,2 - 2,8)(2,2 + 2,8) = (-0,6) \cdot 5 = -3$.

Ответ: -3

4) $3,5^2 - 3,7^2 = (3,5 - 3,7)(3,5 + 3,7) = (-0,2) \cdot 7,2 = -1,44$.

Ответ: -1,44

5) $(\frac{5}{6})^2 - (\frac{2}{3})^2 = (\frac{5}{6} - \frac{2}{3})(\frac{5}{6} + \frac{2}{3}) = (\frac{5}{6} - \frac{4}{6})(\frac{5}{6} + \frac{4}{6}) = \frac{1}{6} \cdot \frac{9}{6} = \frac{1}{6} \cdot \frac{3}{2} = \frac{3}{12} = \frac{1}{4}$.

Ответ: $\frac{1}{4}$

6) $(\frac{7}{9})^2 - (\frac{1}{6})^2 = (\frac{7}{9} - \frac{1}{6})(\frac{7}{9} + \frac{1}{6}) = (\frac{14}{18} - \frac{3}{18})(\frac{14}{18} + \frac{3}{18}) = \frac{11}{18} \cdot \frac{17}{18} = \frac{187}{324}$.

Ответ: $\frac{187}{324}$

7) $(\frac{5}{12})^2 - (\frac{3}{4})^2 = (\frac{5}{12} - \frac{3}{4})(\frac{5}{12} + \frac{3}{4}) = (\frac{5}{12} - \frac{9}{12})(\frac{5}{12} + \frac{9}{12}) = (-\frac{4}{12}) \cdot \frac{14}{12} = (-\frac{1}{3}) \cdot \frac{7}{6} = -\frac{7}{18}$.

Ответ: $-\frac{7}{18}$

8) $(\frac{3}{10})^2 - (\frac{4}{5})^2 = (\frac{3}{10} - \frac{4}{5})(\frac{3}{10} + \frac{4}{5}) = (\frac{3}{10} - \frac{8}{10})(\frac{3}{10} + \frac{8}{10}) = (-\frac{5}{10}) \cdot \frac{11}{10} = (-\frac{1}{2}) \cdot \frac{11}{10} = -\frac{11}{20}$.

Ответ: $-\frac{11}{20}$

9) $(\frac{8}{15})^2 - (\frac{4}{5})^2 = (\frac{8}{15} - \frac{4}{5})(\frac{8}{15} + \frac{4}{5}) = (\frac{8}{15} - \frac{12}{15})(\frac{8}{15} + \frac{12}{15}) = (-\frac{4}{15}) \cdot \frac{20}{15} = (-\frac{4}{15}) \cdot \frac{4}{3} = -\frac{16}{45}$.

Ответ: $-\frac{16}{45}$

10) $(2\frac{1}{7})^2 - (2\frac{1}{7})^2 = (2\frac{1}{7} - 2\frac{1}{7})(2\frac{1}{7} + 2\frac{1}{7}) = 0 \cdot 4\frac{2}{7} = 0$.

Ответ: 0

11) $(3\frac{1}{3})^2 - (4\frac{1}{2})^2 = (\frac{10}{3})^2 - (\frac{9}{2})^2 = (\frac{10}{3} - \frac{9}{2})(\frac{10}{3} + \frac{9}{2}) = (\frac{20}{6} - \frac{27}{6})(\frac{20}{6} + \frac{27}{6}) = (-\frac{7}{6}) \cdot \frac{47}{6} = -\frac{329}{36} = -9\frac{5}{36}$.

Ответ: $-9\frac{5}{36}$

12) $(5\frac{1}{6})^2 - (7\frac{1}{3})^2 = (\frac{31}{6})^2 - (\frac{22}{3})^2 = (\frac{31}{6} - \frac{22}{3})(\frac{31}{6} + \frac{22}{3}) = (\frac{31}{6} - \frac{44}{6})(\frac{31}{6} + \frac{44}{6}) = (-\frac{13}{6}) \cdot \frac{75}{6} = (-\frac{13}{6}) \cdot \frac{25}{2} = -\frac{325}{12} = -27\frac{1}{12}$.

Ответ: $-27\frac{1}{12}$

13) $51^2 - 41^2 = (51 - 41)(51 + 41) = 10 \cdot 92 = 920$.

Ответ: 920

14) $54^2 - 46^2 = (54 - 46)(54 + 46) = 8 \cdot 100 = 800$.

Ответ: 800

15) $76^2 - 24^2 = (76 - 24)(76 + 24) = 52 \cdot 100 = 5200$.

Ответ: 5200

16) $328^2 - 172^2 = (328 - 172)(328 + 172) = 156 \cdot 500 = 78000$.

Ответ: 78000

17) $(3\frac{2}{3})^2 - (2\frac{1}{3})^2 = (\frac{11}{3})^2 - (\frac{7}{3})^2 = (\frac{11}{3} - \frac{7}{3})(\frac{11}{3} + \frac{7}{3}) = \frac{4}{3} \cdot \frac{18}{3} = \frac{4}{3} \cdot 6 = 8$.

Ответ: 8

18) $(7\frac{5}{9})^2 - (4\frac{4}{9})^2 = (\frac{68}{9})^2 - (\frac{40}{9})^2 = (\frac{68}{9} - \frac{40}{9})(\frac{68}{9} + \frac{40}{9}) = \frac{28}{9} \cdot \frac{108}{9} = \frac{28}{9} \cdot 12 = \frac{28 \cdot 4}{3} = \frac{112}{3} = 37\frac{1}{3}$.

Ответ: $37\frac{1}{3}$