Упр.114 ГДЗ Алимов 10-11 класс (Алгебра)

1. Упростить выражение.

1)

$$\frac{\sqrt{x}-\sqrt{y}}{\sqrt[4]{x}-\sqrt[4]{y}}-\frac{\sqrt{x}+\sqrt[4]{xy}}{\sqrt[4]{x}+\sqrt[4]{y}} = \frac{x^{1/2}-y^{1/2}}{x^{1/4}-y^{1/4}}-\frac{x^{1/2}+x^{1/4}y^{1/4}}{x^{1/4}+y^{1/4}}$$
$$= \frac{(x^{1/4}-y^{1/4})(x^{1/4}+y^{1/4})}{x^{1/4}-y^{1/4}} — \frac{x^{1/4}(x^{1/4}+y^{1/4})}{x^{1/4}+y^{1/4}} = x^{1/4}+y^{1/4}-x^{1/4} = y^{1/4}.$$

Ответ: $$\sqrt[4]{y}$$

2)

$$\frac{x-y}{\sqrt[3]{x}-\sqrt[3]{y}}-\frac{x+y}{\sqrt[3]{x}+\sqrt[3]{y}} = \frac{(x-y)(\sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2})}{x-y} — \frac{(x+y)(\sqrt[3]{x^2}-\sqrt[3]{xy}+\sqrt[3]{y^2})}{x+y}$$
$$= \sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} — \sqrt[3]{x^2}+\sqrt[3]{xy}-\sqrt[3]{y^2} = 2\sqrt[3]{xy}.$$

Ответ: $$2\sqrt[3]{xy}$$

3)

$$\frac{\sqrt{x}-\sqrt[3]{y^2}}{\sqrt[4]{x}+\sqrt[3]{y}}+\sqrt[3]{y} = \frac{x^{1/2}-y^{2/3}}{x^{1/4}+y^{1/3}}+y^{1/3} = \frac{(x^{1/4}-y^{1/3})(x^{1/4}+y^{1/3})}{x^{1/4}+y^{1/3}}+y^{1/3}$$
$$= x^{1/4}-y^{1/3}+y^{1/3} = x^{1/4}.$$

Ответ: $$\sqrt[4]{x}$$

4)

$$\frac{x\sqrt{x}-y\sqrt{y}}{x\sqrt{y}-y\sqrt{x}}-1 = \frac{(\sqrt{x})^3-(\sqrt{y})^3}{\sqrt{xy}(\sqrt{x}-\sqrt{y})}-1 = \frac{(\sqrt{x}-\sqrt{y})(x+\sqrt{xy}+y)}{\sqrt{xy}(\sqrt{x}-\sqrt{y})}-1$$
$$= \frac{x+\sqrt{xy}+y}{\sqrt{xy}}-1 = \frac{x+\sqrt{xy}+y-\sqrt{xy}}{\sqrt{xy}} = \frac{x+y}{\sqrt{xy}}.$$

Ответ: $$\frac{x+y}{\sqrt{xy}}$$