Упр.261 Повторение ГДЗ Мерзляк 11 класс Базовый уровень (Алгебра)

Сократите дробь:

1. $$\frac{x-8x^{3/7}}{x^{4/7}-8}$$;
2. $$\frac{5y^{2/3}}{y^{5/6}+y^{2/3}}$$;
3. $$\frac{a^{0,5}+b^{0,5}}{a-b}$$;
4. $$\frac{m^{1,5}-n^{1,5}}{m+m^{0,5}n^{0,5}+n}$$;
5. $$\frac{a-2a^{0,5}b^{0,5}+b}{ab^{0,5}-a^{0,5}b}$$;
6. $$\frac{8a+1}{4a^{2/3}-1}$$;
7. $$\frac{x^{5/8}+6x^{1/4}}{x-36x^{1/4}}$$;
8. $$\frac{26^{1/5}+2^{1/5}}{52^{1/5}+4^{1/5}}$$.

1. $$\frac{x-8x^{3/7}}{x^{4/7}-8}=\frac{x^{3/7}\left(x^{4/7}-8\right)}{x^{4/7}-8}=x^{3/7}.$$

2. $$\frac{5y^{2/3}}{y^{5/6}+y^{2/3}}=\frac{5y^{2/3}}{y^{2/3}\left(y^{1/6}+1\right)}=\frac{5}{y^{1/6}+1}.$$

3. $$\frac{a^{0,5}+b^{0,5}}{a-b}=\frac{a^{0,5}+b^{0,5}}{\left(a^{0,5}-b^{0,5}\right)\left(a^{0,5}+b^{0,5}\right)}=\frac{1}{a^{0,5}-b^{0,5}}=\frac{1}{\sqrt a-\sqrt b}.$$

4. $$\frac{m^{1,5}-n^{1,5}}{m+m^{0,5}n^{0,5}+n}=\frac{\left(m^{0,5}-n^{0,5}\right)\left(m+m^{0,5}n^{0,5}+n\right)}{m+m^{0,5}n^{0,5}+n}=\sqrt m-\sqrt n.$$

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

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

7. $$\frac{x^{5/8}+6x^{1/4}}{x-36x^{1/4}}=\frac{x^{1/4}\left(x^{3/8}+6\right)}{x^{1/4}\left(x^{3/4}-36\right)}=\frac{x^{3/8}+6}{x^{3/4}-36}.$$

   $$x^{3/4}-36=\left(x^{3/8}+6\right)\left(x^{3/8}-6\right),$$

   поэтому
   $$\frac{x^{3/8}+6}{x^{3/4}-36}=\frac{1}{x^{3/8}-6}.$$

8. $$\frac{26^{1/5}+2^{1/5}}{52^{1/5}+4^{1/5}}=\frac{26^{1/5}+2^{1/5}}{2^{1/5}\left(26^{1/5}+2^{1/5}\right)}=\frac{1}{2^{1/5}}=2^{-1/5}.$$

Ответ

1) $$x^{3/7}$$; 2) $$\frac{5}{y^{1/6}+1}$$; 3) $$\frac{1}{\sqrt a-\sqrt b}$$; 4) $$\sqrt m-\sqrt n$$; 5) $$\frac{\sqrt a-\sqrt b}{\sqrt{ab}}$$; 6) $$\frac{4a^{2/3}-2a^{1/3}+1}{2a^{1/3}-1}$$; 7) $$\frac{1}{x^{3/8}-6}$$; 8) $$2^{-1/5}$$.