Упр.74 ГДЗ Колягин Ткачёва 9 класс (Алгебра)

1) (a-b)/(a^(1/3)-b^(1/3))-(a+b)/(a^(1/3)+b^(1/3));
2) (a+b)/(a^(2/3)-a^(1/3)b^(1/3)+b^(2/3))-(a-b)/(a^(2/3)+a^(1/3)b^(1/3)+b^(2/3));
3) (a^(2/3)+b^(2/3))/(a-b)-1/(a^(1/3)-b^(1/3));
4) (a^(1/3)-b^(1/3))/(a+b)+1/(a^(2/3)-a^(1/3)b^(1/3)+b^(2/3)).

1. $$\frac{a-b}{a^{1/3}-b^{1/3}}-\frac{a+b}{a^{1/3}+b^{1/3}}$$
   Представим числители как разность и сумму кубов:
   $$a-b=\left(a^{1/3}-b^{1/3}\right)\left(a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}\right),$$
   $$a+b=\left(a^{1/3}+b^{1/3}\right)\left(a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}\right).$$
   Тогда
   $$\frac{a-b}{a^{1/3}-b^{1/3}}-\frac{a+b}{a^{1/3}+b^{1/3}}$$
   $$=\left(a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}\right)-\left(a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}\right)$$
   $$=2a^{1/3}b^{1/3}=2\sqrt[3]{ab}.$$

2. $$\frac{a+b}{a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}}-\frac{a-b}{a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}}$$
   Используем разложение:
   $$a+b=\left(a^{1/3}+b^{1/3}\right)\left(a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}\right),$$
   $$a-b=\left(a^{1/3}-b^{1/3}\right)\left(a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}\right).$$
   Тогда
   $$\frac{a+b}{a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}}-\frac{a-b}{a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}}$$
   $$=\left(a^{1/3}+b^{1/3}\right)-\left(a^{1/3}-b^{1/3}\right)$$
   $$=2b^{1/3}=2\sqrt[3]{b}.$$

3. $$\frac{a^{2/3}+b^{2/3}}{a-b}-\frac{1}{a^{1/3}-b^{1/3}}$$
   Приведём к общему знаменателю:
   $$a-b=\left(a^{1/3}-b^{1/3}\right)\left(a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}\right).$$
   Тогда
   $$\frac{a^{2/3}+b^{2/3}}{a-b}-\frac{1}{a^{1/3}-b^{1/3}}$$
   $$=\frac{a^{2/3}+b^{2/3}-\left(a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}\right)}{\left(a^{1/3}-b^{1/3}\right)\left(a^{2/3}+a^{1/3}b^{1/3}+b^{2/3}\right)}$$
   $$=\frac{-a^{1/3}b^{1/3}}{a-b}=\frac{\sqrt[3]{ab}}{b-a}.$$

4. $$\frac{a^{1/3}-b^{1/3}}{a+b}+\frac{1}{a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}}$$
   Используем формулу:
   $$a+b=\left(a^{1/3}+b^{1/3}\right)\left(a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}\right).$$
   Тогда
   $$\frac{a^{1/3}-b^{1/3}}{a+b}+\frac{1}{a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}}$$
   $$=\frac{a^{1/3}-b^{1/3}}{\left(a^{1/3}+b^{1/3}\right)\left(a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}\right)}+\frac{a^{1/3}+b^{1/3}}{\left(a^{1/3}+b^{1/3}\right)\left(a^{2/3}-a^{1/3}b^{1/3}+b^{2/3}\right)}$$
   $$=\frac{2a^{1/3}}{a+b}=\frac{2\sqrt[3]{a}}{a+b}.$$

Ответ

1) $$2\sqrt[3]{ab}$$;
2) $$2\sqrt[3]{b}$$;
3) $$\frac{\sqrt[3]{ab}}{b-a}$$;
4) $$\frac{2\sqrt[3]{a}}{a+b}$$.