Упр.13.11 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) (a-4b)/(a^0,5+2b^0,5);
2) (a-b)/(ab^(1/2)+a^(1/2)b);
3) (4c^(2/3)-12c^(1/3)d^(1/3)+9d^(2/3))/(2c^(1/3)-3d^(1/3));
4) (m^(1/2)-n^(1/2))/(m^(3/2)-n^(3/2);
5) (a^(3/4)+7a^(1/2))/(a-49a^(1/2));
6) (30^(1/5)-6^(1/5))/(10^(1/5)-2^(1/5)).

1. $$\frac{a-4b}{a^{0,5}+2b^{0,5}}=\frac{(a^{0,5}-2b^{0,5})(a^{0,5}+2b^{0,5})}{a^{0,5}+2b^{0,5}}=a^{0,5}-2b^{0,5}.$$

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

3. $$\frac{4c^{2/3}-12c^{1/3}d^{1/3}+9d^{2/3}}{2c^{1/3}-3d^{1/3}}=\frac{(2c^{1/3}-3d^{1/3})^2}{2c^{1/3}-3d^{1/3}}=2c^{1/3}-3d^{1/3}.$$

4. $$\frac{m^{1/2}-n^{1/2}}{m^{3/2}-n^{3/2}}=\frac{m^{1/2}-n^{1/2}}{(m^{1/2}-n^{1/2})(m+n^{1/2}m^{1/2}+n)}=\frac{1}{m+m^{1/2}n^{1/2}+n}.$$

5. $$\frac{a^{3/4}+7a^{1/2}}{a-49a^{1/2}}=\frac{a^{1/4}(a^{1/2}+7a^{1/4})}{(a^{1/2}-7a^{1/4})(a^{1/2}+7a^{1/4})}=\frac{a^{1/4}}{a^{1/2}-7a^{1/4}}=\frac{1}{a^{1/4}-7}.$$

6. $$\frac{30^{1/5}-6^{1/5}}{10^{1/5}-2^{1/5}}=\frac{6^{1/5}(5^{1/5}-1)}{2^{1/5}(5^{1/5}-1)}=\frac{6^{1/5}}{2^{1/5}}=3^{1/5}=\sqrt[5]{3}.$$

Ответ

1) $$a^{0,5}-2b^{0,5}$$; 2) $$\frac{1}{b^{1/2}}-\frac{1}{a^{1/2}}$$; 3) $$2c^{1/3}-3d^{1/3}$$; 4) $$\frac{1}{m+m^{1/2}n^{1/2}+n}$$; 5) $$\frac{1}{a^{1/4}-7}$$; 6) $$\sqrt[5]{3}$$.