829 Вправа 829Алгебра 8 клас Бевз 2025 · § 16 · стор. 156

829. Скороти дріб.

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

б) ${\displaystyle \frac{\sqrt{x}+\sqrt{y}}{x-y}=}$${\displaystyle \frac{\cancel{\sqrt{x}+\sqrt{y}}}{(\sqrt{x}-\sqrt{y})\cancel{\left(\sqrt{x}+\sqrt{y}\right)}}=}$${\displaystyle \frac{1}{\sqrt{x}-\sqrt{y}}}$;

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

г) ${\displaystyle \frac{\sqrt{m}-2}{m-2\sqrt{m}}=}$${\displaystyle \frac{\cancel{\sqrt{m}-2}}{\sqrt{m}\cancel{\left(\sqrt{m}-2\right)}}=}$${\displaystyle \frac{1}{\sqrt{m}}}$;

ґ) ${\displaystyle \frac{a\sqrt{b}+b\sqrt{a}}{\sqrt{a}+\sqrt{b}}=}$${\displaystyle \frac{\sqrt{ab}\cancel{\left(\sqrt{a}+\sqrt{b}\right)}}{\cancel{\sqrt{a}+\sqrt{b}}}=}$${\displaystyle \sqrt{ab}}$;

д) ${\displaystyle \frac{a-2\sqrt{3a}+3}{a-\sqrt{3a}}=}$${\displaystyle \frac{{(\sqrt{a}-\sqrt{3})}^2}{\sqrt{a}(\sqrt{a}-\sqrt{3})}=}$${\displaystyle \frac{\sqrt{a}-\sqrt{3}}{\sqrt{a}}}$;

е) ${\displaystyle \frac{m-4\sqrt{m}+4}{m-4}=}$${\displaystyle \frac{{(\sqrt{m}-2)}^2}{(\sqrt{m}-2)(\sqrt{m}+2)}=}$${\displaystyle \frac{\sqrt{m}-2}{\sqrt{m}+2}}$;

є) ${\displaystyle \frac{{(\sqrt{5}-3)}^2}{7-3\sqrt{5}}=}$${\displaystyle \frac{14-6\sqrt{5}}{7-3\sqrt{5}}=}$${\displaystyle \frac{2\cancel{\left(7-3\sqrt{5}\right)}}{\cancel{7-3\sqrt{5}}}=}$${\displaystyle 2}$;

ж) ${\displaystyle \frac{{(\sqrt{6}-2)}^2}{2\sqrt{6}-5}=}$${\displaystyle \frac{10-4\sqrt{6}}{2\sqrt{6}-5}=}$${\displaystyle \frac{-2\cancel{\left(2\sqrt{6}-5\right)}}{\cancel{2\sqrt{6}-5}}=}$${\displaystyle -2}$.