Упр.565 ГДЗ Мерзляк Полонский 8 класс (Алгебра)

1. $$\left(\sqrt{7}+3\right)\left(3\sqrt{7}-1\right)=3\cdot 7-\sqrt{7}+9\sqrt{7}-3=21-3+8\sqrt{7}=18+8\sqrt{7}.$$

2. $$\left(4\sqrt{2}-\sqrt{3}\right)\left(2\sqrt{2}+5\sqrt{3}\right)=4\cdot 2\cdot 2+4\cdot 5\sqrt{6}-2\sqrt{6}-5\cdot 3=16+20\sqrt{6}-2\sqrt{6}-15=1+18\sqrt{6}.$$

3. $$\left(\sqrt{p}-q\right)\left(\sqrt{p}+q\right)=p-q^2.$$

4. $$\left(6-\sqrt{13}\right)\left(6+\sqrt{13}\right)=36-13=23.$$

5. $$\left(\sqrt{5}-x\right)\left(\sqrt{5}+x\right)=5-x^2.$$

6. $$\left(\sqrt{19}+\sqrt{17}\right)\left(\sqrt{19}-\sqrt{17}\right)=19-17=2.$$

7. $$\left(\sqrt{6}+\sqrt{2}\right)^2=6+2\sqrt{12}+2=8+2\cdot 2\sqrt{3}=8+4\sqrt{3}.$$

8. $$\left(3-2\sqrt{15}\right)^2=9-2\cdot 3\cdot 2\sqrt{15}+4\cdot 15=9-12\sqrt{15}+60=69-12\sqrt{15}.$$

Ответ

1. $$18+8\sqrt{7}$$
2. $$1+18\sqrt{6}$$
3. $$p-q^2$$
4. $$23$$
5. $$5-x^2$$
6. $$2$$
7. $$8+4\sqrt{3}$$
8. $$69-12\sqrt{15}$$

1. $$a\sqrt{3}=\sqrt{3a^2}.$$

2. $$b\sqrt{-b}=\sqrt{-b^3}.$$

3. $$c\sqrt{c^5}=\sqrt{c^7}.$$

4. При $$m\ge 0$$:
   $$m\sqrt{n}=\sqrt{m^2n}.$$

5. При $$x\le 0$$:
   $$xy^2\sqrt{xy}=-\sqrt{x^3y^5}.$$

6. $$2p\sqrt{\frac{p}{2}}=\sqrt{4p^2\cdot \frac{p}{2}}=\sqrt{2p^3}.$$

7. $$2p\sqrt{-\frac{p}{2}}=\sqrt{4p^2\cdot \left(-\frac{p}{2}\right)}=\sqrt{-2p^3}.$$

8. При $$a\ge 0$$:
   $$ab^2\sqrt{\frac{a}{b}}=\sqrt{a^2b^4\cdot \frac{a}{b}}=\sqrt{a^3b^3}.$$

Ответ

1. $$\sqrt{3a^2}$$
2. $$\sqrt{-b^3}$$
3. $$\sqrt{c^7}$$
4. $$\sqrt{m^2n}$$
5. $$-\sqrt{x^3y^5}$$
6. $$\sqrt{2p^3}$$
7. $$\sqrt{-2p^3}$$
8. $$\sqrt{a^3b^3}$$