Упр.104 Вариант 2 Дидактические материалы ГДЗ Мерзляк Полонский 8 класс (Алгебра)

1) 4/v15; 5) (a+6)/v(a+6); 9) (x+6)/(v(x+10)-2);
2) 8/v2; 6) 1/(v11-1); 10) (x^2-7x)/(v(x-6)+1);
3) 42/(5v7); 7) 14/(v17+v3); 11) (x^2-25)/(2-v(x-1));
4) m^4/(nvm); 8) 15/(v43-v13); 12) y/(v(5+y)+v(4y+5)).

1. $$\frac{4}{\sqrt{15}}=\frac{4\sqrt{15}}{\sqrt{15}\sqrt{15}}=\frac{4\sqrt{15}}{15}$$

2. $$\frac{8}{\sqrt{2}}=\frac{8\sqrt{2}}{\sqrt{2}\sqrt{2}}=\frac{8\sqrt{2}}{2}=4\sqrt{2}$$

3. $$\frac{42}{5\sqrt{7}}=\frac{42\sqrt{7}}{5\sqrt{7}\sqrt{7}}=\frac{42\sqrt{7}}{5\cdot 7}=\frac{6\sqrt{7}}{5}$$

4. $$\frac{m^4}{n\sqrt{m}}=\frac{m^4\sqrt{m}}{nm}=\frac{m^3\sqrt{m}}{n}$$

5. $$\frac{a+6}{\sqrt{a+6}}=\frac{(a+6)\sqrt{a+6}}{a+6}=\sqrt{a+6}$$

6. $$\frac{1}{\sqrt{11}-1}=\frac{\sqrt{11}+1}{(\sqrt{11}-1)(\sqrt{11}+1)}=\frac{\sqrt{11}+1}{11-1}=\frac{\sqrt{11}+1}{10}$$

7. $$\frac{14}{\sqrt{17}+\sqrt{3}}=\frac{14(\sqrt{17}-\sqrt{3})}{(\sqrt{17}+\sqrt{3})(\sqrt{17}-\sqrt{3})}=\frac{14(\sqrt{17}-\sqrt{3})}{17-3}=\sqrt{17}-\sqrt{3}$$

8. $$\frac{15}{\sqrt{43}-\sqrt{13}}=\frac{15(\sqrt{43}+\sqrt{13})}{(\sqrt{43}-\sqrt{13})(\sqrt{43}+\sqrt{13})}=\frac{15(\sqrt{43}+\sqrt{13})}{43-13}=\frac{\sqrt{43}+\sqrt{13}}{2}$$

9. $$\frac{x+6}{\sqrt{x+10}-2}=\frac{(x+6)(\sqrt{x+10}+2)}{(\sqrt{x+10}-2)(\sqrt{x+10}+2)}$$
   $$=\frac{(x+6)(\sqrt{x+10}+2)}{x+10-4}=\frac{(x+6)(\sqrt{x+10}+2)}{x+6}=\sqrt{x+10}+2$$

10. $$\frac{x^2-7x}{\sqrt{x-6}+1}=\frac{(x^2-7x)(\sqrt{x-6}-1)}{(\sqrt{x-6}+1)(\sqrt{x-6}-1)}$$
    $$=\frac{(x^2-7x)(\sqrt{x-6}-1)}{x-6-1}=\frac{x(x-7)(\sqrt{x-6}-1)}{x-7}=x(\sqrt{x-6}-1)$$

11. $$\frac{x^2-25}{2-\sqrt{x-1}}=\frac{(x^2-25)(2+\sqrt{x-1})}{(2-\sqrt{x-1})(2+\sqrt{x-1})}$$
    $$=\frac{(x-5)(x+5)(2+\sqrt{x-1})}{4-(x-1)}=\frac{(x-5)(x+5)(2+\sqrt{x-1})}{5-x}$$
    $$=-(x+5)(2+\sqrt{x-1})$$

12. $$\frac{y}{\sqrt{5+y}+\sqrt{4y+5}}=\frac{y(\sqrt{5+y}-\sqrt{4y+5})}{(\sqrt{5+y}+\sqrt{4y+5})(\sqrt{5+y}-\sqrt{4y+5})}$$
    $$=\frac{y(\sqrt{5+y}-\sqrt{4y+5})}{5+y-(4y+5)}=\frac{y(\sqrt{5+y}-\sqrt{4y+5})}{-3y}$$
    $$=-\frac{\sqrt{5+y}-\sqrt{4y+5}}{3}=\frac{\sqrt{4y+5}-\sqrt{5+y}}{3}$$

Ответ

1) $$\frac{4\sqrt{15}}{15}$$; 2) $$4\sqrt{2}$$; 3) $$\frac{6\sqrt{7}}{5}$$; 4) $$\frac{m^3\sqrt{m}}{n}$$; 5) $$\sqrt{a+6}$$; 6) $$\frac{\sqrt{11}+1}{10}$$; 7) $$\sqrt{17}-\sqrt{3}$$; 8) $$\frac{\sqrt{43}+\sqrt{13}}{2}$$; 9) $$\sqrt{x+10}+2$$; 10) $$x(\sqrt{x-6}-1)$$; 11) $$-(x+5)(2+\sqrt{x-1})$$; 12) $$\frac{\sqrt{4y+5}-\sqrt{5+y}}{3}$$.