Упр.10.23 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) $$\frac{a-5a^{1/2}}{a^{1/2}-5}=\frac{a^{1/2}\left(a^{1/2}-5\right)}{a^{1/2}-5}=a^{1/2}.$$

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

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

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

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

6) $$\frac{a+b}{a^{1/3}+b^{1/3}}=\frac{a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}}{1}\cdot\frac{a^{1/3}+b^{1/3}}{a^{1/3}+b^{1/3}}=a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}.$$

7) $$\frac{m^{1/2}-n^{1/2}}{m^{3/2}-n^{3/2}}=\frac{m^{1/2}-n^{1/2}}{\left(m^{1/2}-n^{1/2}\right)\left(m+m^{1/2}n^{1/2}+n\right)}=\frac{1}{m+\left(mn\right)^{1/2}+n}.$$

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

9) $$\frac{30^{1/5}-6^{1/5}}{10^{1/5}-2^{1/5}}=\frac{6^{1/5}\left(5^{1/5}-1\right)}{2^{1/5}\left(5^{1/5}-1\right)}=\frac{6^{1/5}}{2^{1/5}}=3^{1/5}.$$

Ответ: 1) $$a^{1/2}$$; 2) $$a^{0,5}-2b^{0,5}$$; 3) $$b^{-1/2}-a^{-1/2}$$; 4) $$a^{1/2}+b^{1/2}$$; 5) $$2c^{1/3}-3d^{1/3}$$; 6) $$a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}$$; 7) $$\frac{1}{m+\left(mn\right)^{1/2}+n}$$; 8) $$\frac{1}{a^{1/4}-7}$$; 9) $$3^{1/5}$$.