Упр.561 ГДЗ Мерзляк Полонский 8 класс (Алгебра)

1. Упростите выражение:

   1) $$4\sqrt{700}-27\sqrt{7}$$;

   2) $$\sqrt{75}-6\sqrt{3}$$;

   3) $$2\sqrt{50}-8\sqrt{2}$$;

   4) $$5\sqrt{12}-7\sqrt{3}$$;

   5) $$3\sqrt{72}-4\sqrt{2}+2\sqrt{98}$$;

   6) $$\frac{1}{3}\sqrt{108}+\sqrt{363}-\frac{2}{9}\sqrt{243}$$.

2. Упростите выражение:

   1) $$\frac{a}{\sqrt{a}-2}-\frac{4\sqrt{a}-4}{\sqrt{a}-2}$$;

   2) $$\frac{\sqrt{m}+1}{\sqrt{m}-2}-\frac{\sqrt{\ }+3}{\sqrt{m}}$$.

1. $$4\sqrt{700}-27\sqrt7=4\sqrt{7\cdot100}-27\sqrt7=4\cdot10\sqrt7-27\sqrt7=40\sqrt7-27\sqrt7=13\sqrt7.$$
2. $$\sqrt{75}-6\sqrt3=\sqrt{25\cdot3}-6\sqrt3=5\sqrt3-6\sqrt3=-\sqrt3.$$
3. $$2\sqrt{50}-8\sqrt2=2\sqrt{25\cdot2}-8\sqrt2=2\cdot5\sqrt2-8\sqrt2=10\sqrt2-8\sqrt2=2\sqrt2.$$
4. $$5\sqrt{12}-7\sqrt3=5\sqrt{4\cdot3}-7\sqrt3=5\cdot2\sqrt3-7\sqrt3=10\sqrt3-7\sqrt3=3\sqrt3.$$
5. $$3\sqrt{72}-4\sqrt2+2\sqrt{98}=3\sqrt{36\cdot2}-4\sqrt2+2\sqrt{49\cdot2}=3\cdot6\sqrt2-4\sqrt2+2\cdot7\sqrt2$$
   $$=18\sqrt2-4\sqrt2+14\sqrt2=28\sqrt2.$$
6. $$\frac13\sqrt{108}+\sqrt{363}-\frac29\sqrt{243}=\frac13\sqrt{36\cdot3}+\sqrt{121\cdot3}-\frac29\sqrt{81\cdot3}$$
   $$=\frac13\cdot6\sqrt3+11\sqrt3-\frac29\cdot9\sqrt3=2\sqrt3+11\sqrt3-2\sqrt3=11\sqrt3.$$
7. $$\frac{a}{\sqrt a-2}-\frac{4\sqrt a-4}{\sqrt a-2}=\frac{a-4\sqrt a+4}{\sqrt a-2}=\frac{(\sqrt a-2)^2}{\sqrt a-2}=\sqrt a-2.$$
8. $$\frac{\sqrt m+1}{\sqrt m-2}-\frac{\sqrt m+3}{\sqrt m}= \frac{\sqrt m(\sqrt m+1)-(\sqrt m+3)(\sqrt m-2)}{\sqrt m(\sqrt m-2)}$$
   $$=\frac{m+\sqrt m-\left(m-2\sqrt m+3\sqrt m-6\right)}{\sqrt m(\sqrt m-2)}= \frac{m+\sqrt m-m-\sqrt m+6}{\sqrt m(\sqrt m-2)}$$
   $$=\frac{6}{\sqrt m(\sqrt m-2)}=\frac{6}{m-2\sqrt m}.$$

Ответ

1. $$13\sqrt7$$
2. $$-\sqrt3$$
3. $$2\sqrt2$$
4. $$3\sqrt3$$
5. $$28\sqrt2$$
6. $$11\sqrt3$$
7. $$\sqrt a-2$$
8. $$\frac{6}{m-2\sqrt m}$$