276 Вправа 276Алгебра 8 клас Бевз 2025 · § 6 · стор. 59

276. Simplify the expression.

a) ${\displaystyle \frac{-5a^2}{4c^3}:(\frac{a^3}{2c^3}:\frac{a}{c^2})}$;

1) ${\displaystyle \frac{a^3}{2c^3}:\frac{a}{c^2}=}$${\displaystyle \frac{a^3}{2c^3}\cdot \frac{c^2}{a}=}$${\displaystyle \frac{a^3\cdot c^2}{2c^3\cdot a}=}$${\displaystyle \frac{a^2\cancel{a}\cdot \cancel{c^2}}{2c\cancel{c^2}\cdot \cancel{a}}=}$${\displaystyle \frac{a^2}{2c}}$;

2) ${\displaystyle \frac{-5a^2}{4c^3}:\frac{a^2}{2c}=}$${\displaystyle \frac{-5a^2}{4c^3}\cdot \frac{2c}{a^2}=}$${\displaystyle \frac{-5a^2\cdot 2c}{4c^3\cdot a^2}}$=

=${\displaystyle -\frac{5\cancel{a^2}\cdot \cancel{2}\cancel{c}}{2\cdot \cancel{2}{c}^2\cancel{c}\cdot \cancel{a^2}}=}$${\displaystyle -\frac{5}{2c^2}}$;

b) ${\displaystyle \frac{-9c^2}{5a}:{(\frac{3c^2}{5a^3}\cdot \frac{a}{c})}^3}$;

1) ${\displaystyle \frac{3c^2}{5a^3}\cdot \frac{a}{c}=}$${\displaystyle \frac{3c^2\cdot a}{5a^3\cdot c}=}$${\displaystyle \frac{3c\cancel{c}\cdot \cancel{a}}{5a^2\cancel{a}\cdot \cancel{c}}=}$${\displaystyle \frac{3c}{5a^2}}$;

2) ${\displaystyle {\left(\frac{3c}{5a^2}\right)}^3=}$${\displaystyle \frac{27c^3}{125a^6}}$;

3) ${\displaystyle \frac{-9c^2}{5a}:\frac{27c^3}{125a^6}=}$${\displaystyle \frac{-9c^2}{5a}\cdot \frac{125a^6}{27c^3}=}$${\displaystyle \frac{-9c^2\cdot 125a^6}{5a\cdot 27c^3}}$=

=${\displaystyle -\frac{\cancel{9}\cancel{c^2}\cdot 25\cdot \cancel{5}{a}^5\cancel{a}}{\cancel{5}\cancel{a}\cdot 3\cdot \cancel{9}c\cancel{c^2}}=}$${\displaystyle -\frac{25a^5}{3c}}$;

c) ${\displaystyle \frac{8mn}{9a{x}^2}:(\frac{6m}{5x}:\frac{3x^2}{2n})}$;

1) ${\displaystyle \frac{6m}{5x}:\frac{3x^2}{2n}=}$${\displaystyle \frac{6m}{5x}\cdot \frac{2n}{3x^2}=}$${\displaystyle \frac{6m\cdot 2n}{5x\cdot 3x^2}=}$${\displaystyle \frac{2\cdot \cancel{3}m\cdot 2n}{5x\cdot \cancel{3}{x}^2}=}$${\displaystyle \frac{4mn}{5x^3}}$;

2) ${\displaystyle \frac{8mn}{9a{x}^2}:\frac{4mn}{5x^3}=}$${\displaystyle \frac{8mn}{9a{x}^2}\cdot \frac{5x^3}{4mn}=}$${\displaystyle \frac{8mn\cdot 5x^3}{9a{x}^2\cdot 4mn}}$=

=${\displaystyle \frac{2\cdot \cancel{4}\cancel{m}\cancel{n}\cdot 5x\cancel{x^2}}{9a\cancel{x^2}\cdot \cancel{4}\cancel{m}\cancel{n}}=}$${\displaystyle \frac{10x}{9a}}$;

d) ${\displaystyle {\left(\frac{-2a}{9c^{2}x}\right)}^3:{\left(\frac{2a}{3c{x}^2}\right)}^4}$;

1) ${\displaystyle {\left(\frac{-2a}{9c^{2}x}\right)}^3=}$${\displaystyle -\frac{8a^3}{729c^6{x}^3}}$;

2) ${\displaystyle {\left(\frac{2a}{3c{x}^2}\right)}^4=}$${\displaystyle \frac{16a^4}{81c^4{x}^8}}$;

3) ${\displaystyle \frac{-8a^3}{729c^6{x}^3}:\frac{16a^4}{81c^4{x}^8}=}$${\displaystyle \frac{-8a^3}{729c^6{x}^3}\cdot \frac{81c^4{x}^8}{16a^4}}$=

=${\displaystyle \frac{-8a^3\cdot 81c^4{x}^8}{729c^6{x}^3\cdot 16a^4}=}$${\displaystyle -\frac{\cancel{8}\cancel{a^3}\cdot \cancel{81}\cancel{c^4}{x}^5\cancel{x^3}}{9\cdot \cancel{81}{c}^2\cancel{c^4}\cancel{x^3}\cdot 2\cdot \cancel{8}a\cancel{a^3}}=}$${\displaystyle -\frac{x^5}{18a{c}^2}}$;

e) ${\displaystyle (\frac{0,5ax}{2m}:\frac{x}{4m^2}):\frac{m}{a}}$;

1) ${\displaystyle \frac{0,5ax}{2m}:\frac{x}{4m^2}=}$${\displaystyle \frac{0,5ax}{2m}\cdot \frac{4m^2}{x}=}$${\displaystyle \frac{0,5ax\cdot 4m^2}{2m\cdot x}}$=

=${\displaystyle \frac{0,5\cdot 2\cdot \cancel{2}a\cancel{x}m\cancel{m}}{\cancel{2}\cancel{m}\cancel{x}}=}$${\displaystyle am}$;

2) ${\displaystyle am:\frac{m}{a}=}$${\displaystyle am\cdot \frac{a}{m}=}$${\displaystyle \frac{a\cancel{m}\cdot a}{\cancel{m}}=}$${\displaystyle a^2}$.