Решение №17129: \(\frac{a\left ( a-2 \right )-b\left ( b+2 \right )+\sqrt{ab}\left ( b-a+2 \right )}{a+b-\sqrt{ab}}:\left ( 1+2\frac{a^{2}+b^{2}+ab}{b^{3}-a^{3}} \right )=\frac{a^{2}-2a-b^{2}-2b-\sqrt{ab}\left ( a-b-2 \right )}{a+b-\sqrt{ab}}:\left ( 1-2\frac{a^{2}+b^{2}+ab}{b^{3}-a^{3}} \right )=\frac{\left ( a-b \right )\left ( a+b \right )-2\left ( a+b \right )-\sqrt{ab}\left ( a-b-2 \right )}{a+b-\sqrt{ab}}:\left ( 1-\frac{2}{a-b} \right )=\frac{\left ( a-b-2 \right )\left ( a+b-\sqrt{ab} \right )}{a+b-\sqrt{ab}}\cdot \frac{a-b}{a-b-2}=a-b\)

Ответ: \(a-b\)

Решение №17130: \(\left ( \sqrt[3]{m^{2}}+n\sqrt[3]{m}+n^{2} \right )\cdot \frac{\sqrt[3]{m^{4}}-n^{3}+n^{2}\sqrt[3]{m}-mn}{mn^{-1}+n-n^{4}m^{-1}-n^{2}}=\left ( m^{\frac{2}{3}}+nm^{\frac{1}{3}}+n^{2} \right )\frac{m^{\frac{4}{3}}-n^{3}+n^{2}m^{\frac{1}{3}}-mn}{\frac{m}{n}+n-\frac{n^{4}}{m}-n^{2}}=\left ( m^{\frac{2}{3}}+nm^{\frac{1}{3}}+n^{2} \right )\frac{\left ( m+n^{2} \right )\left ( m^{\frac{1}{3}}-n \right )}{\frac{\left ( m+n^{2} \right )\left ( m-n^{3} \right )}{mn}}=\frac{\left ( m-n^{3} \right )mn}{m-n^{3}}=mn\)

Ответ: \(mn\)

Решение №17131: \(\left ( \frac{pq^{2}}{\left ( p+q \right )^{\frac{5}{2}}}-\frac{2pq^{2}}{\left ( p+q \right )^{\frac{3}{2}}}+\frac{pq}{\sqrt{p+q}} \right ):\left ( \frac{p^{2}}{\left ( p+q \right )^{\frac{5}{2}}}-\frac{p^{2}q}{\left ( p+q \right )^{\frac{7}{2}}} \right )=\frac{pq}{\left ( p+q \right )^{\frac{1}{2}}}\cdot \left ( \frac{q^{2}}{\left ( p+q \right )^{2}} -\frac{2q}{p+q}+1\right ):\frac{p^{2}}{\left ( p+q \right )^{\frac{5}{2}}}\cdot \left ( 1-\frac{q}{p+q} \right )=\frac{pq}{\left ( p+q \right )^{\frac{1}{2}}}\left ( \frac{q^{2}-2q\left ( p+q \right )+\left ( p+q \right )^{2}}{\left ( p+q \right )^{2}} \right ):\frac{p^{2}}{\left ( p+q \right )^{\frac{5}{2}}}\left ( \frac{p+q-q}{p+q} \right )=\frac{p^{3}q}{\left ( p+q \right )^{\frac{5}{2}}}\cdot \frac{\left ( p+q \right )^{\frac{5}{2}}\left ( p+q \right )}{p^{3}}=q\left ( p+q \right )\)

Ответ: \(q\left ( p+q \right )\)

Решение №17132: \(\frac{\sqrt{1+\sqrt{1-x^{2}}}\left ( \sqrt{\left ( 1+x \right )^{3}}-\sqrt{\left ( 1-x \right )^{3}} \right )}{2+\sqrt{1-x^{2}}}=\sqrt{1+\sqrt{1-x^{2}}}\left ( \sqrt{1+x}-\sqrt{1-x} \right )=\sqrt{\frac{1+x+2\sqrt{1-x^{2}+1-x}}{2}}\left ( \sqrt{1+x}-\sqrt{1-x} \right )=\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{2}}\left ( \sqrt{1+x}-\sqrt{1-x} \right )=\frac{1+x-1+x}{\sqrt{2}}=\frac{2x}{\sqrt{2}}=x\sqrt{2}\)

Ответ: \(x\sqrt{2}\)

Решение №17133: \(x\sqrt[3]{2x\sqrt{xy}-x\sqrt{3xy}}\cdot \sqrt[6]{x^{2}y\left ( 7+4\sqrt{3} \right )}=x\sqrt[3]{x\sqrt{xy\left ( 2-\sqrt{3} \right )}}\cdot \sqrt[6]{x^{3}y\left ( 7+4\sqrt{3} \right )}=x\sqrt[3]{x\sqrt{xy}\left ( 2-\sqrt{3} \right )}\cdot \sqrt[6]{\left | x \right |\sqrt{xy}\left ( 2+\sqrt{3} \right )}=x\sqrt[3]{x\left | x \right |\left | x \right |\left | y \right |}=x\sqrt[3]{x^{3}\left | y \right |}=x\cdot x\cdot \left | \sqrt[3]{y} \right |=x^{2}\left | \sqrt[3]{y} \right |\)

Ответ: \(x^{2}\left | \sqrt[3]{y} \right |\)

Решение №17134: \(\frac{x^{4}+x^{2}+x\sqrt{2}+2}{x^{2}-x\sqrt{2}+2}-x\sqrt{2}=\frac{x^{4}+x^{2}+x\sqrt{2}+2-x^{3}\sqrt{2}+2x^{2}-2x\sqrt{2}}{x^{2}-x\sqrt{2}+2}=\frac{x^{4}-\sqrt{2}x^{3}+3x^{2}-\sqrt{2}x+2}{x^{2}-x\sqrt{2}+2}=\frac{\left ( x^{2}-\sqrt{2}x+2 \right )\left ( x^{2}+1 \right )}{x^{2}-x\sqrt{2}+2}=x^{2}+1\)

Ответ: \(x^{2}+1\)

Решение №17135: \(\frac{\left ( \sqrt{a}+\sqrt{ax}+x+x\sqrt{x} \right )^{2}\left ( 1-\sqrt{x} \right )^{2}}{\left ( x+x^{-1}-2 \right )a^{-\frac{1}{4}}}-\frac{\left ( x\sqrt{a} \right )^{\frac{3}{2}}}{\left ( ax^{-1}+4\sqrt{a}+4x \right )^{-\frac{1}{2}}}=\frac{\left ( \sqrt{a}\left ( 1+\sqrt{x} \right ) +x\left ( 1+\sqrt{x} \right )\right )^{2}\left ( 1-\sqrt{x} \right )^{2}}{\left ( x+\frac{1}{x}-2 \right )\frac{1}{\sqrt[4]{a}}}-\frac{\sqrt{x^{3}}\sqrt[4]{a^{3}}}{\left ( \frac{a}{x}+4\sqrt{a}+4x \right )^{-\frac{1}{2}}}=\left ( \sqrt{a}+x \right )^{2}\cdot x\sqrt[4]{a}-x\sqrt[4]{a^{3}}\left ( \sqrt{a}+2x \right )=x\sqrt[4]{a}\left ( \left ( \sqrt{a}+x \right )^{2}-\sqrt[4]{a^{2}}\left ( \sqrt{a}+2x \right ) \right )=x\sqrt[4]{a}\left ( a+2\sqrt{a}\cdot x+x^{2}-a-2\sqrt{a}\cdot x \right )=x\sqrt[4]{a}\cdot x^{2}=x^{3}\cdot \sqrt[4]{a}\)

Ответ: \(x^{3}\cdot \sqrt[4]{a}\)

Решение №17136: \(\frac{x-1}{x+x^{\frac{1}{2}}+1}:\frac{x^{0.5}+1}{x^{1.5}-1}+\frac{2}{x^{-0.5}}=\frac{\left ( x^{\frac{1}{2}}-1 \right )\left ( x^{\frac{1}{2}}+1 \right )}{x+x^{\frac{1}{2}}+1}\cdot \frac{\left ( x^{\frac{1}{2}} \right )^{3}-1}{x^{\frac{1}{2}}+1}+\frac{2}{\frac{1}{x^{\frac{1}{2}}}}=\frac{\left ( x^{\frac{1}{2}}-1 \right )\left ( x^{\frac{1}{2}}+1 \right )\left (x+x^{\frac{1}{2}}+1 \right )}{x+x^{\frac{1}{2}}+1}+2x^{\frac{1}{2}}=\left ( x^{\frac{1}{2}}-1 \right )^{2}+2x^{\frac{1}{2}}=x-2x\frac{1}{2}+1+2x\frac{1}{2}=x+1\)

Ответ: \(x+1\)

Решение №17137: \(\frac{\sqrt[4]{x^{5}}+\sqrt[4]{xy^{4}}-\sqrt[4]{x^{4}y}-\sqrt[4]{x^{5}}}{\sqrt{x}-\sqrt{y}}\cdot \left ( \sqrt[4]{x}+\sqrt[4]{y} \right )=\frac{\sqrt[4]{x}\left ( \sqrt[4]{x^{4}}+\sqrt[4]{y^{4}} \right )-\sqrt[4]{y}\left ( \sqrt[4]{x^{4}}+\sqrt[4]{y^{4}} \right )}{\left ( \sqrt[4]{x}-\sqrt[4]{y} \right )\left ( \sqrt[4]{x}+\sqrt[4]{y} \right )}\cdot \left ( \sqrt[4]{x}+\sqrt[4]{y} \right )=\sqrt[4]{x^{4}}+\sqrt[4]{y^{4}}=x+y\)

Ответ: \(x+y\)

Решение №17138: \(\left ( \left ( \frac{2^{\frac{3}{2}}+27y^{\frac{3}{5}}}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt[10]{32y^{2}}-2 \right )\cdot 3^{-2} \right )^{5}=\left ( \left ( \frac{\left ( \sqrt{2} \right )^{3}+\left ( 3\sqrt[5]{y} \right )^{3}}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt{2}\sqrt[5]{y}-2 \right )\cdot \frac{1}{9} \right )^{5}=\left ( \left ( \frac{\left ( \sqrt{2}+3\sqrt[5]{y} \right )\left ( \left ( \sqrt{2} \right )^{2}-3\sqrt{2}\sqrt[5]{y}+\left ( 3\sqrt[5]{y} \right )^{2} \right )}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt{2}\sqrt[5]{y}-2 \right ) \cdot \frac{1}{9}\right )^{5}=\left ( 9\sqrt[5]{y} \cdot \frac{1}{9}\right )^{5}=\left ( \sqrt[5]{y} \right )^{5}=y^{2}\)

Ответ: \(y^{2}\)