Упр.23.17 ГДЗ Мерзляк Поляков 9 класс (Алгебра)

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

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

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

   $$\frac{a^{1/2}-b^{1/2}}{a^{1/2}b^{1/2}}=\frac{1}{b^{1/2}}-\frac{1}{a^{1/2}}.$$

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

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

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

   $$a+b=(a^{1/3}+b^{1/3})(a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}),$$

   значит,
   $$\frac{a+b}{a^{1/3}+b^{1/3}}=a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}.$$

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

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

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

   $$a^{1/2}-49=(a^{1/4}-7)(a^{1/4}+7),$$

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

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

Ответ

1) $$a^{0,5}$$; 2) $$a^{0,5}-2b^{0,5}$$; 3) $$b^{-0,5}-a^{-0,5}$$; 4) $$a^{0,5}+b^{0,5}$$; 5) $$2c^{1/3}-3d^{1/3}$$; 6) $$a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}$$; 7) $$\frac{1}{m+n+(mn)^{0,5}}$$; 8) $$\frac{1}{a^{0,25}-7}$$; 9) $$3^{1/5}$$.