187 Вправа 187Алгебра 8 клас Бевз 2025 · § 4 · стор. 40

187. Спрости вираз.

а) ${\displaystyle \frac{a^2-8}{a^2-16}-\frac{a-3}{a-4}=}$${\displaystyle \frac{a^2-8}{(a-4)(a+4)}-\frac{{a-3}^{\mathrm{\setminus }a+4}}{a-4}}$=

=${\displaystyle \frac{a^2-8-(a-3)(a+4)}{(a-4)(a+4)}=}$${\displaystyle \frac{a^2-8-a^2-a+12}{(a-4)(a+4)}}$=

=${\displaystyle \frac{4-a}{(a-4)(a+4)}=}$${\displaystyle \frac{-\cancel{\left(a-4\right)}}{(a+4)\cancel{\left(a-4\right)}}=}$${\displaystyle -\frac{1}{a+4}}$;

б) ${\displaystyle \frac{2}{a^2-9}-\frac{1}{a^2+3a}=}$${\displaystyle \frac{2^{\mathrm{\setminus }a}}{(a-3)(a+3)}-\frac{1^{\mathrm{\setminus }a-3}}{a(a+3)}}$=

=${\displaystyle \frac{2a-(a-3)}{a(a-3)(a+3)}=}$${\displaystyle \frac{2a-a+3}{a(a-3)(a+3)}}$=

=${\displaystyle \frac{\cancel{\left(a+3\right)}}{a(a-3)\cancel{\left(a+3\right)}}=}$${\displaystyle \frac{1}{a\left(a-3\right)}}$;

в) ${\displaystyle \frac{a+2b}{2a-b}-\frac{6ab+2b^2}{4a^2-b^2}=}$${\displaystyle \frac{{a+2b}^{\mathrm{\setminus }2a+b}}{2a-b}-\frac{6ab+2b^2}{(2a-b)(2a+b)}}$=

=${\displaystyle \frac{(a+2b)(2a+b)-(6ab+2b^2)}{(2a-b)(2a+b)}}$=

=${\displaystyle \frac{2a^2+5ab+2b^2-6ab-2b^2}{(2a-b)(2a+b)}=}$${\displaystyle \frac{2a^2-ab}{(2a-b)(2a+b)}}$=

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

г) ${\displaystyle \frac{c}{c-2}+\frac{2c-4}{c^2-4c+4}=}$${\displaystyle \frac{c^{\mathrm{\setminus }c-2}}{c-2}+\frac{2c-4}{{(c-2)}^2}}$=

=${\displaystyle \frac{c(c-2)+2c-4}{{(c-2)}^2}=}$${\displaystyle \frac{c^2-2c+2c-4}{{(c-2)}^2}=}$${\displaystyle \frac{c^2-4}{{(c-2)}^2}}$=

=${\displaystyle \frac{(c+2)\cancel{\left(c-2\right)}}{(c-2)\cancel{\left(c-2\right)}}=}$${\displaystyle \frac{c+2}{c-2}}$.