Упр.13.12 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) (a+2a^(1/3))/(a^(2/3)+2);
2) (a-b^2)/(a-a^(1/2)b);
3) (x^3,5y^2,5-x^2,5y^3,5)/(x+2x^0,5y^0,5+y);
4) (a-125)/(a^(2/3)-25);
5) (m^(7/6)-36m^(5/6))/(m^(1/2)-6m^(1/3));
6) (24^(1/4)-8^(1/4))/(6^(1/4)-2^(1/4)).

1. $$\frac{a+2a^{1/3}}{a^{2/3}+2}=\frac{a^{1/3}\left(a^{2/3}+2\right)}{a^{2/3}+2}=a^{1/3}.$$

2. $$\frac{a-b^2}{a-a^{1/2}b}=\frac{\left(a^{1/2}-b\right)\left(a^{1/2}+b\right)}{a^{1/2}\left(a^{1/2}-b\right)}=\frac{a^{1/2}+b}{a^{1/2}}=1+\frac{b}{a^{1/2}}.$$

3. $$\frac{x^{3.5}y^{2.5}-x^{2.5}y^{3.5}}{x+2x^{0.5}y^{0.5}+y} =\frac{x^{2.5}y^{2.5}(x-y)}{\left(x^{0.5}+y^{0.5}\right)^2} =x^{2.5}y^{2.5}\cdot\frac{x^{0.5}-y^{0.5}}{x^{0.5}+y^{0.5}}.$$

4. $$\frac{a-125}{a^{2/3}-25} =\frac{\left(a^{1/3}-5\right)\left(a^{2/3}+5a^{1/3}+25\right)}{\left(a^{1/3}-5\right)\left(a^{1/3}+5\right)} =\frac{a^{2/3}+5a^{1/3}+25}{a^{1/3}+5}.$$

5. $$\frac{m^{7/6}-36m^{5/6}}{m^{1/2}-6m^{1/3}} =\frac{m^{5/6}\left(m^{1/3}-36\right)}{m^{1/3}\left(m^{1/6}-6\right)} =m^{1/2}\cdot\frac{m^{1/3}-36}{m^{1/6}-6}.$$

6. $$\frac{24^{1/4}-8^{1/4}}{6^{1/4}-2^{1/4}} =\frac{2^{1/4}\left(3^{1/4}-1\right)}{2^{1/4}\left(3^{1/4}-1\right)} =2^{1/4}.$$

Ответ

1) $$a^{1/3}$$; 2) $$1+\frac{b}{a^{1/2}}$$; 3) $$x^{2.5}y^{2.5}\cdot\frac{x^{0.5}-y^{0.5}}{x^{0.5}+y^{0.5}}$$; 4) $$\frac{a^{2/3}+5a^{1/3}+25}{a^{1/3}+5}$$; 5) $$m^{1/2}\cdot\frac{m^{1/3}-36}{m^{1/6}-6}$$; 6) $$2^{1/4}$$.