Упр.117 ГДЗ Алимов 10-11 класс (Алгебра)

1)

$$\left(\frac{(\sqrt[4]{a}+\sqrt[4]{b})^2+(\sqrt[4]{a}-\sqrt[4]{b})^2}{a+\sqrt{ab}}\right)^5\cdot \sqrt[3]{a^{10}\cdot \sqrt a}$$
$$=\left(\frac{(a^{1/4}+b^{1/4})^2+(a^{1/4}-b^{1/4})^2}{a+(ab)^{1/2}}\right)^5\cdot \left(a^{10}\cdot a^{1/2}\right)^{1/3}$$
$$=\left(\frac{2a^{1/2}+2b^{1/2}}{a+a^{1/2}b^{1/2}}\right)^5\cdot a^{21/6} =\left(\frac{2(a^{1/2}+b^{1/2})}{a^{1/2}(a^{1/2}+b^{1/2})}\right)^5\cdot a^{7/2}$$
$$=\left(\frac{2}{a^{1/2}}\right)^5\cdot a^{7/2} =\frac{2^5}{a^{5/2}}\cdot a^{7/2} =32a.$$

Ответ: $$32a$$

2)

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

Ответ: $$a^{-1}$$

3)

$$\left(\frac{a^{3/2}-b^{3/2}}{\sqrt a-\sqrt b} -\sqrt{\frac{ab\sqrt[3]{a}+ab^{4/3}}{a^{1/3}+b^{1/3}}}\right)\cdot \frac1{a+b}$$
$$=\left(\frac{(\sqrt a-\sqrt b)(a+\sqrt{ab}+b)}{\sqrt a-\sqrt b} -\sqrt{ab\cdot \frac{\sqrt[3]{a}+\sqrt[3]{b}}{a^{1/3}+b^{1/3}}}\right)\cdot \frac1{a+b}$$
$$=\left(a+\sqrt{ab}+b-\sqrt{ab}\right)\cdot \frac1{a+b} =\frac{a+b}{a+b} =1.$$

Ответ: $$1$$