Упр.72 ГДЗ Колягин Ткачёва 9 класс (Алгебра)

1) (1-2v(b/a)+b/a):(a^(1/2)-b^(1/2))^2;
2) (a^(1/3)+b^(1/3)):(2+(a/b)^(1/3)+(b/a)^(1/3));
3) (a^(1/4)-a^(9/4))/(a^(1/4)-a^(5/4))-(b^(-1/2)-b^(3/2))/(b^(1/2)+b^(-1/2));
4) (va-a^(-1/2) b)/(1-v(a^(-1)b))-((a^2)^(1/3)-a^(-1/3)b)/(a^(1/6)+a^(-1/3)vb).

1. $$\left(1-2\sqrt{\frac{b}{a}}+\frac{b}{a}\right):\left(\sqrt{a}-\sqrt{b}\right)^2$$

   Заметим, что
   $$1-2\sqrt{\frac{b}{a}}+\frac{b}{a}=\left(1-\sqrt{\frac{b}{a}}\right)^2.$$
   Тогда
   $$\left(1-\sqrt{\frac{b}{a}}\right)^2:\left(\sqrt{a}-\sqrt{b}\right)^2.$$
   Так как
   $$1-\sqrt{\frac{b}{a}}=\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}},$$
   получаем
   $$\left(\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}}\right)^2:\left(\sqrt{a}-\sqrt{b}\right)^2=\frac{1}{(\sqrt{a})^2}=\frac{1}{a}.$$

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

   Преобразуем знаменатель:
   $$2+\left(\frac{a}{b}\right)^{1/3}+\left(\frac{b}{a}\right)^{1/3} =\left(\frac{a^{1/6}}{b^{1/6}}+\frac{b^{1/6}}{a^{1/6}}\right)^2.$$
   Тогда
   $$\left(a^{1/3}+b^{1/3}\right):\left(2+\left(\frac{a}{b}\right)^{1/3}+\left(\frac{b}{a}\right)^{1/3}\right) =\frac{a^{1/3}+b^{1/3}}{\left(\frac{a^{1/6}}{b^{1/6}}+\frac{b^{1/6}}{a^{1/6}}\right)^2}.$$
   После преобразований получаем
   $$\frac{\sqrt[3]{ab}}{\sqrt[3]{a}+\sqrt[3]{b}}.$$

3. $$\frac{a^{1/4}-a^{9/4}}{a^{1/4}-a^{5/4}}-\frac{b^{-1/2}-b^{3/2}}{b^{1/2}+b^{-1/2}}$$

   Вынесем общие множители:
   $$\frac{a^{1/4}(1-a^2)}{a^{1/4}(1-a)}-\frac{b^{-1/2}(1-b^2)}{b^{-1/2}(b+1)}.$$
   Тогда
   $$\frac{1-a^2}{1-a}-\frac{1-b^2}{1+b}.$$
   Разложим на множители:
   $$\frac{(1-a)(1+a)}{1-a}-\frac{(1-b)(1+b)}{1+b}=1+a-(1-b)=a+b.$$

4. $$\frac{\sqrt{a}-a^{-1/2}b}{1-\sqrt{a^{-1}b}}-\frac{(a^2)^{1/3}-a^{-1/3}b}{a^{1/6}+a^{-1/3}b}$$

   Преобразуем каждую дробь:
   $$\frac{a^{1/2}(1-a^{-1}b)}{1-a^{-1/2}b^{1/2}}-\frac{a^{-1/3}(a-b)}{a^{-1/3}(a^{1/2}+b^{1/2})}.$$
   Получаем
   $$\frac{a^{1/2}(1-a^{-1/2}b^{1/2})(1+a^{-1/2}b^{1/2})}{1-a^{-1/2}b^{1/2}}-\frac{a^{1/2}-b^{1/2}}{a^{1/2}+b^{1/2}}.$$
   После сокращения:
   $$a^{1/2}\left(1+a^{-1/2}b^{1/2}\right)-\left(a^{1/2}-b^{1/2}\right)=2b^{1/2}.$$

Ответ

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