654 Вправа 654Алгебра 8 клас Тарасенкова 2025 · § 15 · стор. 193

654. Спростіть вираз:

1) ${\displaystyle \frac{{\sqrt{5}}^{\mathrm{\setminus }\left(\sqrt{a}-\sqrt{5}\right)}}{\sqrt{a}+\sqrt{5}}-\frac{{\sqrt{5}}^{\mathrm{\setminus }\left(\sqrt{a}+\sqrt{5}\right)}}{\sqrt{a}-\sqrt{5}}=}$${\displaystyle \frac{\sqrt{5}(\sqrt{a}-\sqrt{5})-\sqrt{5}(\sqrt{a}+\sqrt{5})}{{\left(\sqrt{a}\right)}^2-{\left(\sqrt{5}\right)}^2}=}$${\displaystyle \frac{\sqrt{5a}-5-(\sqrt{5a}+5)}{a-5}=}$${\displaystyle \frac{\sqrt{5a}-5-\sqrt{5a}-5}{a-5}=}$${\displaystyle \frac{-10}{a-5}=}$${\displaystyle \frac{10}{5-a}}$;

2) ${\displaystyle \frac{\sqrt{m}}{(\sqrt{m}-5)(\sqrt{m}+5)}-\frac{1^{\mathrm{\setminus }\left(\sqrt{m}+5\right)}}{\sqrt{m}-5}=}$${\displaystyle \frac{\sqrt{m}-(\sqrt{m}+5)}{m-25}=}$${\displaystyle \frac{\sqrt{m}-\sqrt{m}-5}{m-25}=}$${\displaystyle \frac{-5}{m-25}=}$${\displaystyle \frac{5}{25-m}}$;

3) ${\displaystyle \frac{\sqrt{11}}{(\sqrt{m}-\sqrt{11})(\sqrt{m}+\sqrt{11})}-\frac{1^{\mathrm{\setminus }\left(\sqrt{m}+\sqrt{11}\right)}}{\sqrt{m}-\sqrt{11}}=}$${\displaystyle \frac{\sqrt{11}-(\sqrt{m}+\sqrt{11})}{m-11}=}$${\displaystyle \frac{\sqrt{11}-\sqrt{m}-\sqrt{11}}{m-11}=}$${\displaystyle \frac{-\sqrt{m}}{m-11}=}$${\displaystyle \frac{\sqrt{m}}{11-m}}$;

4) ${\displaystyle \frac{\sqrt{5p}}{(\sqrt{p}-\sqrt{5})(\sqrt{p}+\sqrt{5})}+\frac{{\sqrt{p}}^{\mathrm{\setminus }\left(\sqrt{p}-\sqrt{5}\right)}}{\sqrt{p}+\sqrt{5}}=}$${\displaystyle \frac{\sqrt{5p}+\sqrt{p}(\sqrt{p}-\sqrt{5})}{p-5}=}$${\displaystyle \frac{\sqrt{5p}+p-\sqrt{5p}}{p-5}=}$${\displaystyle \frac{p}{p-5}}$.