73 Вправа 73Алгебра 8 клас Бевз 2025 · § 2 · стор. 20

73. Доведи тотожність.

а) ${\displaystyle \frac{6ab}{2b^{2}c}=}$${\displaystyle \frac{3ac}{b{c}^2}}$;

${\displaystyle \frac{6ab}{2b^{2}c}=}$${\displaystyle \frac{3a\cdot \cancel{2b}}{\cancel{2b}\cdot bc}=}$${\displaystyle \frac{3a}{bc}}$;

${\displaystyle \frac{3ac}{b{c}^2}=}$${\displaystyle \frac{3a\cdot \cancel{c}}{bc\cdot \cancel{c}}=}$${\displaystyle \frac{3a}{bc}}$;

${\displaystyle \frac{3a}{bc}=}$${\displaystyle \frac{3a}{bc}}$ — тотожність доведено;

б) ${\displaystyle \frac{a^2-1}{a+1}=}$${\displaystyle a-1}$;

${\displaystyle \frac{a^2-1}{a+1}=}$${\displaystyle \frac{(a-1)\cancel{\left(a+1\right)}}{\cancel{a+1}}=}$${\displaystyle a-1}$;

${\displaystyle a-1=}$${\displaystyle a-1}$ — тотожність доведено;

в) ${\displaystyle \frac{2(x-y)}{xy-y^2}=}$${\displaystyle \frac{4(x-y)}{2y(x-y)}}$;

${\displaystyle \frac{2(x-y)}{xy-y^2}=}$${\displaystyle \frac{2\cancel{\left(x-y\right)}}{y\cancel{\left(x-y\right)}}=}$${\displaystyle \frac{2}{y}}$;

${\displaystyle \frac{4(x-y)}{2y(x-y)}=}$${\displaystyle \frac{2\cdot \cancel{2}\cancel{\left(x-y\right)}}{\cancel{2}y\cancel{\left(x-y\right)}}=}$${\displaystyle \frac{2}{y}}$;

${\displaystyle \frac{2}{y}=}$${\displaystyle \frac{2}{y}}$ — тотожність доведено;

г) ${\displaystyle \frac{4a^2-x^2}{2a+x}=}$${\displaystyle 2a-x}$;

${\displaystyle \frac{4a^2-x^2}{2a+x}=}$${\displaystyle \frac{(2a-x)\cancel{\left(2a+x\right)}}{\cancel{2a+x}}=}$${\displaystyle 2a-x}$;

${\displaystyle 2a-x=}$${\displaystyle 2a-x}$ — тотожність доведено;

ґ) ${\displaystyle \frac{{(a+c)}^3}{a+c}=}$${\displaystyle {(a+c)}^2}$;

${\displaystyle \frac{{(a+c)}^3}{a+c}=}$${\displaystyle \frac{{(a+c)}^2\cancel{\left(a+c\right)}}{\cancel{a+c}}=}$${\displaystyle {(a+c)}^2}$;

${\displaystyle {(a+c)}^2=}$${\displaystyle {(a+c)}^2}$ — тотожність доведено;

д) ${\displaystyle \frac{x^2-2x+1}{x-1}=}$${\displaystyle x-1}$;

${\displaystyle \frac{x^2-2x+1}{x-1}=}$${\displaystyle \frac{(x-1)\cancel{\left(x-1\right)}}{\cancel{x-1}}=}$${\displaystyle x-1}$;

${\displaystyle x-1=}$${\displaystyle x-1}$ — тотожність доведено.