Задачи

Фильтрация

Показать фильтрацию

По классам:

По предметам:

По подготовке:

По классам:

По авторам:

Упростить выражение и вычислить \(\sqrt[4]{\left ( 1-2a+a^{2} \right )\left ( a^{2}-1 \right )\left ( a-1 \right )}:\frac{a^{2}+2a-3}{\sqrt[4]{a+1}}\)

Решение №16927: \(\sqrt[4]{\left ( 1-2a+a^{2} \right )\left ( a^{2}-1 \right )\left ( a-1 \right )}:\frac{a^{2}+2a-3}{\sqrt[4]{a+1}}=\sqrt[4]{\left ( a-1 \right )^{2}\left ( a-1 \right )\left ( a+1 \right )\left ( a-1 \right )}\cdot \frac{\sqrt[4]{a+1}}{a^{2}+2a-3}=\frac{\sqrt[4]{\left ( a-1 \right )^{4}\left ( a+1 \right )^{2}}}{\left ( a-1 \right )\left ( a+3 \right )}=\frac{\left | a-1 \right |\sqrt[4]{\left ( a+1 \right )^{2}}}{\left ( a-1 \right )\left ( a+3 \right )}=\frac{\left | a-1 \right |\sqrt{\left ( a+1 \right )}}{\left ( a-1 \right )\left ( a+3 \right )}=-\frac{\sqrt{a+1}}{a+3};\frac{\sqrt{a+1}}{a+3}\)

Ответ: \(-\frac{\sqrt{a+1}}{a+3};\frac{\sqrt{a+1}}{a+3}\)

Упростить выражение и вычислить \(\frac{\sqrt{2a+2\sqrt{a^{2}-b^{2}}}-\sqrt{a-b}}{\sqrt{2a-2\sqrt{a^{2}-b^{2}}}+\sqrt{a-b}}\)

Решение №16928: \(\frac{\sqrt{2a+2\sqrt{a^{2}-b^{2}}}-\sqrt{a-b}}{\sqrt{2a-2\sqrt{a^{2}-b^{2}}}+\sqrt{a-b}}=\frac{\sqrt{a+b}+\sqrt{a-b}-\sqrt{a-b}}{\left | \sqrt{a+b}-\sqrt{a-b} \right |+\sqrt{a-b}}=\frac{\sqrt{a+b}}{\left | \sqrt{a+b}-\sqrt{a-b} \right |+\sqrt{a-b}}=\frac{\sqrt{a+b}}{2\sqrt{a-b}-\sqrt{a+b}};\frac{\sqrt{a+b}}{\sqrt{a+b}}=\frac{\sqrt{a+b}}{2\sqrt{a-b}-\sqrt{a+b}};1\)

Ответ: \(\frac{\sqrt{a+b}}{2\sqrt{a-b}-\sqrt{a+b}};1\)

Упростить выражение \(\frac{\left ( m-1 \right )\sqrt{m}-\left ( n-1 \right )\sqrt{n}}{\sqrt{m^{3}n}+mn+m^{2}-m}\)

Решение №16929: \(\frac{\left ( m-1 \right )\sqrt{m}-\left ( n-1 \right )\sqrt{n}}{\sqrt{m^{3}n}+mn+m^{2}-m}=\frac{m\sqrt{m}-\sqrt{m}-n\sqrt{n}+\sqrt{n}}{m\sqrt{mn}+mn+m^{2}-m}=\frac{\left ( \sqrt{m^{3}}-\sqrt{n^{3}} \right )-\left ( \sqrt{m}-\sqrt{n} \right )}{m\left ( \sqrt{mn}+n+m-1 \right )}=\frac{\left ( \sqrt{m}-\sqrt{n} \right )\left ( m+\sqrt{mn}+n-1 \right )}{m\left ( \sqrt{mn}+n+m-1 \right )}=\frac{\sqrt{m}-\sqrt{n}}{m}\)

Ответ: \(\frac{\sqrt{m}-\sqrt{n}}{m}\)

Упростить выражение и вычислить \(\frac{\sqrt{\frac{p^{4}+q^{4}}{p^{4}-p^{2}q^{2}}+\frac{2q^{2}}{p^{2}-q^{2}}\left ( p^{3}-pq^{2} \right )-2q\sqrt{p}}}{\sqrt{\frac{p}{p-q}}-\frac{q}{p+q}-\frac{2pq}{p^{2}-q^{2}}\cdot \left ( p-q \right )}\)

Решение №16930: \(\frac{\sqrt{\frac{p^{4}+q^{4}}{p^{4}-p^{2}q^{2}}+\frac{2q^{2}}{p^{2}-q^{2}}\left ( p^{3}-pq^{2} \right )-2q\sqrt{p}}}{\sqrt{\frac{p}{p-q}}-\frac{q}{p+q}-\frac{2pq}{p^{2}-q^{2}}\cdot \left ( p-q \right )}=\frac{p-q}{\sqrt{p}}\sqrt{\frac{p+q}{p-q}}=\frac{1}{\sqrt{p}}\sqrt{\frac{\left ( p-q \right )^{2}\left ( p+q \right )}{p-q}}=\frac{1}{\sqrt{p}}\sqrt{\left ( p-q \right )\left ( p+q \right )}=\frac{\sqrt{p^{2}-q^{2}}}{\sqrt{p}}\)

Ответ: \(\frac{\sqrt{p^{2}-q^{2}}}{\sqrt{p}}\)

Упростить выражение \(\left ( \frac{t\sqrt{t+2}}{\sqrt{t-2}}-\frac{2\sqrt{t-2}}{\sqrt{t+2}}-\frac{4t}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}:\sqrt[4]{t^{2}-4}\)

Решение №16931: \(\left ( \frac{t\sqrt{t+2}}{\sqrt{t-2}}-\frac{2\sqrt{t-2}}{\sqrt{t+2}}-\frac{4t}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}:\sqrt[4]{t^{2}-4}=\left ( \frac{t\left ( t+2 \right )-2\left ( t-2 \right )-4t}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}:\sqrt[4]{t^{2}-4}=\left ( \frac{t^{2}-4t+4}{\sqrt{t^{2}-4}} \right )^{\frac{1}{2}}\cdot \frac{1}{\sqrt[4]{t^{2}-4}}=\frac{t-2}{\sqrt[4]{t^{2}-4}}\cdot \frac{1}{\sqrt[4]{t^{2}-4}}=\frac{t-2}{\sqrt{t^{2}-4}}=\frac{\sqrt{t-2}}{\sqrt{t+2}}=\frac{\sqrt{t^{2}-4}}{t+2}\)

Ответ: \(\frac{\sqrt{t^{2}-4}}{t+2}\)

Упростить выражение и вычислить \(\frac{\sqrt{\sqrt{3}+2}\sqrt[4]{7-4\sqrt{3}}+\sqrt[3]{\sqrt{x}\left ( x+27 \right )-9x-27}}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt[4]{7+4\sqrt{3}}}\)

Решение №16932: \(\frac{\sqrt{\sqrt{3}+2}\sqrt[4]{7-4\sqrt{3}}+\sqrt[3]{\sqrt{x}\left ( x+27 \right )-9x-27}}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt[4]{7+4\sqrt{3}}}=\frac{\sqrt{2+\sqrt{3}}\sqrt[4]{\left ( 2-\sqrt{3} \right )^{2}}+\sqrt{x}-3}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt[4]{\left ( 2+\sqrt{3} \right )^{2}}}=\frac{\sqrt{2+\sqrt{3}}\sqrt{2-\sqrt{3}}+\sqrt{x}-3}{\sqrt{x}-2-\sqrt{2-\sqrt{3}}\sqrt{2+\sqrt{3}}}=\frac{\sqrt{4-3}+\sqrt{x}-3}{\sqrt{x}-2-\sqrt{4-3}}=\frac{1+\sqrt{x}-3}{\sqrt{x}-2-1}=\frac{\sqrt{x}-2}{\sqrt{x}-3}\)

Ответ: \(\frac{\sqrt{x}-2}{\sqrt{x}-3}\)

Упростить выражение \(\left ( \frac{\sqrt[4]{x^{3}}-\sqrt[4]{x}}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}} \right )^{2}\cdot \left ( 1+\frac{2}{\sqrt{x}}+\frac{1}{x} \right )^{-\frac{1}{2}}\)

Решение №16933: \(\left ( \frac{\sqrt[4]{x^{3}}-\sqrt[4]{x}}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}} \right )^{2}\cdot \left ( 1+\frac{2}{\sqrt{x}}+\frac{1}{x} \right )^{-\frac{1}{2}}=\left ( \frac{\sqrt[4]{x}\left ( \sqrt{x}-1 \right )}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}} \right )^{2}\cdot \left (\frac{x}{\left ( \sqrt{x}+1 \right )}\right )^{\frac{1}{2}}=\left ( -\sqrt[4]{x} +\frac{1+\sqrt{x}}{\sqrt[4]{x}}\right )^{2}\cdot \frac{\sqrt{x}}{\sqrt{x}+1}=\frac{1}{\sqrt{x}}\cdot \frac{\sqrt{x}}{\sqrt{x}+1}=\frac{1}{\sqrt{x}+1}=\frac{1-\sqrt{x}}{1-x}\)

Ответ: \(\frac{1-\sqrt{x}}{1-x}\)

Упростить выражение \(\frac{\left ( \sqrt[5]{a^{\frac{4}{3}}} \right )^{\frac{3}{2}}}{\left ( \sqrt[5]{a^{4}} \right )^{3}}:\frac{\left ( \sqrt{a\sqrt[3]{a^{2}b}} \right )^{4}}{\left ( \sqrt[4]{a\sqrt{b}} \right )^{6}}\)

Решение №16934: \(\frac{\left ( \sqrt[5]{a^{\frac{4}{3}}} \right )^{\frac{3}{2}}}{\left ( \sqrt[5]{a^{4}} \right )^{3}}:\frac{\left ( \sqrt{a\sqrt[3]{a^{2}b}} \right )^{4}}{\left ( \sqrt[4]{a\sqrt{b}} \right )^{6}}=\frac{\left ( a^{\frac{4}{15}} \right )^{\frac{3}{2}}}{\left ( a^{\frac{4}{5}} \right )^{3}}\cdot \frac{\left ( \sqrt{a\cdot a^{\frac{2}{3}}b^{\frac{1}{3}}} \right )^{4}}{\left ( \sqrt[4]{ab^{\frac{1}{2}}} \right )^{6}}=\frac{a^{\frac{2}{5}}}{a^{\frac{12}{5}}}\cdot \frac{\left ( a^{\frac{5}{6}}b^{\frac{1}{6}} \right )^{4}}{a^{\frac{6}{4}}b^{\frac{6}{8}}}=a^{-2}\cdot a^{\frac{10}{3}-\frac{3}{2}}\cdot b^{\frac{2}{3}-\frac{3}{4}}=a^{-2}\cdot a^{\frac{11}{6}}\cdot b^{-\frac{1}{12}}=\frac{1}{\sqrt[12]{a^{2}b}}\)

Ответ: \(\frac{1}{\sqrt[12]{a^{2}b}}\)

Упростить выражение и вычислить \(\frac{\sqrt[3]{\sqrt{5}-\sqrt{3}}\sqrt[6]{8+2\sqrt{15}}-\sqrt[3]{a}}{\sqrt[3]{\sqrt{2}+\sqrt{12}}\sqrt[6]{8-2\sqrt{15}}-2\sqrt[3]{2a}+\sqrt[3]{a^{2}}}\)

Решение №16935: \(\frac{\sqrt[3]{\sqrt{5}-\sqrt{3}}\sqrt[6]{8+2\sqrt{15}}-\sqrt[3]{a}}{\sqrt[3]{\sqrt{2}+\sqrt{12}}\sqrt[6]{8-2\sqrt{15}}-2\sqrt[3]{2a}+\sqrt[3]{a^{2}}}=\frac{\sqrt[3]{5-3}-\sqrt[3]{a}}{\sqrt[3]{2\left ( 5-3 \right )}-2\sqrt[3]{2a}+\sqrt[3]{a^{2}}}=\frac{\sqrt[3]{2}-\sqrt[3]{a}}{\left ( \sqrt[3]{2}-\sqrt[3]{a} \right )}=\frac{1}{\sqrt[3]{2}-\sqrt[3]{a}}\)

Ответ: \(\frac{1}{\sqrt[3]{2}-\sqrt[3]{a}}\)

Упростить выражение и вычислить \(\frac{\sqrt{\sqrt{5}-2}\sqrt[4]{9+4\sqrt{5}}+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{\sqrt{\sqrt{5}+2}\sqrt[4]{9-4\sqrt{5}}+a}\)

Решение №16936: \(\frac{\sqrt{\sqrt{5}-2}\sqrt[4]{9+4\sqrt{5}}+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{\sqrt{\sqrt{5}+2}\sqrt[4]{9-4\sqrt{5}}+a}=\frac{\sqrt{\sqrt{5}-2}\sqrt[4]{5+4\sqrt{5}+4}+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{\sqrt{\sqrt{5}+2}\sqrt[4]{5-4\sqrt{5}+4}+a}=\frac{1+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{1+a}=\frac{\sqrt[3]{a^{2}}-\sqrt[3]{a}+1}{\left ( \sqrt[3]{a} \right )^{3}+1}=\frac{1}{\sqrt[3]{a}+1}\)

Ответ: \(\frac{1}{\sqrt[3]{a}+1}\)

Упростить выражение и вычислить \(\frac{1+2a^{\frac{1}{4}}-a^{\frac{1}{2}}}{1-a+4a^{\frac{3}{4}}-4a^{\frac{1}{2}}}+\frac{a^{\frac{1}{4}}-2}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}\)

Решение №16937: \(\frac{1+2a^{\frac{1}{4}}-a^{\frac{1}{2}}}{1-a+4a^{\frac{3}{4}}-4a^{\frac{1}{2}}}+\frac{a^{\frac{1}{4}}-2}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{a^{\frac{2}{4}}-2a^{\frac{1}{4}}-1}{\left ( a^{\frac{1}{4}}-1\left ( a^{\frac{2}{4}}-a^{\frac{2}{4}} \right )-\left ( 2a^{\frac{2}{4}}-2a^{\frac{1}{4}} \right )-\left ( a^{\frac{1}{4}}-1 \right ) \right )}+\frac{a^{\frac{1}{4}}-2}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{1}{( a^{\frac{1}{4}}-1 \right )^{2}}+\frac{a^{\frac{1}{4}}-2}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{1+a^{\frac{1}{4}}-2}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{a^{\frac{1}{4}}-1}{\left ( a^{\frac{1}{4}}-1 \right )^{2}}=\frac{1}{\sqrt[4]{a}-1}\)

Ответ: \(\frac{1}{\sqrt[4]{a}-1}\)

Упростить выражение и вычислить \(\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2}\left ( 2x+\sqrt{x^{2}-1} \right )}{\sqrt{\left ( x+1 \right )^{3}}-\sqrt{\left ( x-1 \right )^{3}}}\)

Решение №16938: \(\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2}\left ( 2x+\sqrt{x^{2}-1} \right )}{\sqrt{\left ( x+1 \right )^{3}}-\sqrt{\left ( x-1 \right )^{3}}}=\frac{\sqrt{\frac{\left ( \sqrt{x+1}-\sqrt{x-1} \right )^{2}}{\sqrt{x^{2}-1}}}}{\sqrt{x+1}-\sqrt{x-1}}=\sqrt{\frac{\left ( \sqrt{x+1}-\sqrt{x-1} \right )^{2}}{\sqrt{x^{2}-1}\left ( \sqrt{x+1}-\sqrt{x-1} \right )^{2}}}=\sqrt{\frac{1}{\sqrt{x^{2}-1}}}=\frac{1}{\sqrt[4]{x^{2}-1}}\)

Ответ: \(\frac{1}{\sqrt[4]{x^{2}-1}}\)

Упростить выражение \(\left ( \frac{a+2}{\sqrt{2a}}-\frac{a}{\sqrt{2a}+2}+\frac{2}{a-\sqrt{2a}} \right )\cdot \frac{\sqrt{a}-\sqrt{2}}{a+2}\)

Решение №16939: \(\left ( \frac{a+2}{\sqrt{2a}}-\frac{a}{\sqrt{2a}+2}+\frac{2}{a-\sqrt{2a}} \right )\cdot \frac{\sqrt{a}-\sqrt{2}}{a+2}=\left ( \frac{a+2}{\sqrt{2a}}-\frac{a}{\sqrt{2}\left ( \sqrt{a}+\sqrt{2} \right )}+\frac{2}{\sqrt{a}\left ( \sqrt{a}-\sqrt{2} \right )} \right )\cdot \frac{\sqrt{a}-\sqrt{2}}{a+2}=\frac{a^{2}-4-a^{2}+a\sqrt{2a}+2\sqrt{2a}+4}{\sqrt{2a}\left ( \sqrt{a}+\sqrt{2} \right )}\cdot \frac{1}{a+2}=\frac{\sqrt{2a}\left ( a+2 \right )}{\sqrt{2a}\left ( \sqrt{a}+\sqrt{2} \right )\left ( a+2 \right )}=\frac{1}{\sqrt{a}+\sqrt{2}}\)

Ответ: \(\frac{1}{\sqrt{a}+\sqrt{2}}\)

Упростить выражение и вычислить \(\left ( \left ( a-3\sqrt[6]{a^{5}}+9\sqrt[3]{a^{2}} \right )\left ( \sqrt{a} +3\sqrt[3]{a}+3\sqrt[12]{a^{5}}\right )^{-1}+3\sqrt[12]{a^{5}} \right )^{-1}\)

Решение №16940: \(\left ( \left ( a-3\sqrt[6]{a^{5}}+9\sqrt[3]{a^{2}} \right )\left ( \sqrt{a} +3\sqrt[3]{a}+3\sqrt[12]{a^{5}}\right )^{-1}+3\sqrt[12]{a^{5}} \right )^{-1}=\left ( \frac{\sqrt[12]{a^{12}}-3\sqrt[12]{a^{10}}+9\sqrt[12]{a^{8}}}{\sqrt[12]{a^{6}}+3\sqrt[12]{a^{4}}+3\sqrt[12]{a^{5}}} +3\sqrt[12]{a^{5}}\right )^{-1}=\left ( \frac{\sqrt[12]{a^{4}}\left ( \sqrt[12]{a^{2}+3\sqrt[12]{a}+3} \right )\left ( \sqrt[12]{a^{2}}+3 \right )}{\sqrt[12]{a^{2}}+3\sqrt[12]{a}+3} \right )^{-1}=\left ( \sqrt[12]{a^{4}}\left ( \sqrt[12]{a^{2}}+3 \right ) \right )^{-1}=\frac{1}{\sqrt[12]{a^{4}}\left ( \sqrt[12]{a^{2}}+3 \right )}=\frac{1}{\sqrt[12]{a^{6}}+3\sqrt[12]{a^{4}}}=\frac{1}{\sqrt{a}+3\sqrt[3]{a}}\)

Ответ: \(\frac{1}{\sqrt{a}+3\sqrt[3]{a}}\)

Упростить выражение и вычислить \(\frac{\sqrt{\frac{1}{a+2\sqrt{a-2}-1}}+\sqrt{\frac{1}{a-2\sqrt{a-2}-1}}}{\sqrt{\frac{1}{a+2\sqrt{a-2}-1}}-\sqrt{\frac{1}{a-2\sqrt{a-2}-1}}}\)

Решение №16941: \(\frac{\sqrt{\frac{1}{a+2\sqrt{a-2}-1}}+\sqrt{\frac{1}{a-2\sqrt{a-2}-1}}}{\sqrt{\frac{1}{a+2\sqrt{a-2}-1}}-\sqrt{\frac{1}{a-2\sqrt{a-2}-1}}}=\frac{\sqrt{\frac{1}{\left ( \sqrt{a-2}+1 \right )^{2}}}+\frac{1}{\left ( \sqrt{a-2}-1 \right )^{2}}}{\frac{1}{\left ( \sqrt{a-2}+1 \right )^{2}}-\frac{1}{\left ( \sqrt{a-2}-1 \right )^{2}}}=\frac{\left | \sqrt{a-2}-1 \right |+\sqrt{a-2}+1}{\left | \sqrt{a-2}-1 \right |-\sqrt{a-2}-1}=-\frac{1}{\sqrt{a-2}};-\sqrt{a-2}\)

Ответ: \(-\frac{1}{\sqrt{a-2}};-\sqrt{a-2}\)

Упростить выражение и вычислить \(\frac{\sqrt{2b+2\sqrt{b^{2}-4}}}{\sqrt{b^{2}-4}+b+2}\)

Решение №16942: \(\frac{\sqrt{2b+2\sqrt{b^{2}-4}}}{\sqrt{b^{2}-4}+b+2}=\frac{\sqrt{b+2\sqrt{\left ( b+2 \right )\left ( b-2 \right )}}+b}{\sqrt{b^{2}-4}+b+2}=\frac{\sqrt{b+2}+\sqrt{b-2}}{\sqrt{b+2}\left ( \sqrt{b-2}+\sqrt{b+2} \right )}=\frac{1}{\sqrt{b+2}}\)

Ответ: \(\frac{1}{\sqrt{b+2}}\)

Упростить выражение \(\frac{4x\left ( x+\sqrt{x^{2}-1} \right )^{2}}{\left ( x+\sqrt{x^{2}-1} \right )^{4}-1}\)

Решение №16943: \(\frac{4x\left ( x+\sqrt{x^{2}-1} \right )^{2}}{\left ( x+\sqrt{x^{2}-1} \right )^{4}-1}=\frac{4x\left ( x+\sqrt{x^{2}-1} \right )^{2}}{\left ( x^{2}+2x\sqrt{x^{2}-1}+x^{2}-1 \right )^{2}-1}=\frac{4x\left ( x+\sqrt{x^{2}-1} \right )^{2}}{\left ( 2x^{2}+2x\sqrt{x^{2}-1}-1 \right )^{2}-1}=\frac{4x\left ( x+\sqrt{x^{2}-1} \right )^{2}}{\left ( 2x^{2}+2x\sqrt{x^{2}-1}-2 \right )\left ( 2x^{2}+2x\sqrt{x^{2} \right )}=\frac{x+\sqrt{x^{2}-1}}{\left ( x^{2}-1 \right )+x\sqrt{x^{2}-1}}=\frac{1}{\sqrt{x^{2}-1}}\)

Ответ: \(\frac{1}{\sqrt{x^{2}-1}}\)

Упростить выражение и вычислить \(\frac{\left | x-1 \right |+\left | x \right |+x}{3x^{2}-4x+1}\)

Решение №16944: \(\frac{\left | x-1 \right |+\left | x \right |+x}{3x^{2}-4x+1}=\frac{-x+1-x+x}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{-x+1+x+x}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{x-1-x+x}{\left ( x-1 \right )\left ( 3x-1 \right )}=\frac{x-1}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{x+1}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{3x-1}{\left ( x-1 \right )\left ( 3x-1 \right )}=\frac{1}{1-3x};\frac{x+1}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{1}{x-1};\)

Ответ: \(\frac{1}{1-3x};\frac{x+1}{\left ( x-1 \right )\left ( 3x-1 \right )};\frac{1}{x-1};\)

Упростить выражение и вычислить \(\frac{\left ( \sqrt[4]{a}+\sqrt[4]{b}-\sqrt[8]{ab} \right )\left ( \sqrt[4]{b}+\sqrt[4]{a}+\sqrt[8]{ab} \right )}{\sqrt[4]{a^{3}b}-b}:\frac{\left ( \sqrt[8]{a}+\sqrt[8]{b} \right )+\left ( \sqrt[8]{a}-\sqrt[8]{b} \right )^{2}}{\left ( \sqrt{a}-\sqrt{b} \right )b^{-\frac{1}{4}}}\)

Решение №16945: \(\frac{\left ( \sqrt[4]{a}+\sqrt[4]{b}-\sqrt[8]{ab} \right )\left ( \sqrt[4]{b}+\sqrt[4]{a}+\sqrt[8]{ab} \right )}{\sqrt[4]{a^{3}b}-b}:\frac{\left ( \sqrt[8]{a}+\sqrt[8]{b} \right )+\left ( \sqrt[8]{a}-\sqrt[8]{b} \right )^{2}}{\left ( \sqrt{a}-\sqrt{b} \right )b^{-\frac{1}{4}}}=\frac{\left ( \sqrt[8]{a^{2}}+\sqrt[8]{b^{2}} \right )^{2}-\left ( \sqrt[8]{ab} \right )^{2}}{\sqrt[8]{b^{2}\left ( \sqrt[8]{a^{2}}-\sqrt[8]{b^{2}} \right )}}:\frac{\left ( 2\sqrt[8]{a^{2}}+2\sqrt[8]{b^{2}} \right )\sqrt[8]{b^{2}}}{\sqrt[8]{a^{4}}-\sqrt[8]{b^{4}}}=\frac{1}{\sqrt[8]{b^{2}}\left ( \sqrt[8]{a^{2}}-\sqrt[8]{b^{2}} \right )}\cdot \frac{\sqrt[8]{a^{2}}-\sqrt[8]{b^{2}} }{2\sqrt[8]{b^{2}}}=\frac{1}{2\sqrt[8]{b^{4}}}=\frac{1}{2\sqrt{b}}\)

Ответ: \(\frac{1}{2\sqrt{b}}\)

Упростить выражение и вычислить \(\frac{\sqrt{1+\left ( \frac{x^{2}-1}{2x} \right )^{2}}}{\left ( x^{2}+1 \right )\frac{1}{x}}\)

Решение №16946: \(\frac{\sqrt{1+\left ( \frac{x^{2}-1}{2x} \right )^{2}}}{\left ( x^{2}+1 \right )\frac{1}{x}}=\frac{\sqrt{\frac{x^{4}+2x^{2}+1}{4x^{2}}}}{\frac{x^{2}+1}{x}}=\frac{\frac{x^{2}+1}{2\left | x \right |}}{\frac{x^{2}+1}{x}}=\frac{x}{2\left | x \right |}=-\frac{1}{2};\frac{1}{2}\)

Ответ: \(-\frac{1}{2};\frac{1}{2}\)

Упростить выражение \(\left ( \frac{4}{a+\frac{1}{b+\frac{1}{c}}}:\frac{1}{a+\frac{1}{b}}-\frac{4}{b\left ( abc+a+c \right )} \right )^{-\frac{1}{2}}\)

Решение №16947: \(\left ( \frac{4}{a+\frac{1}{b+\frac{1}{c}}}:\frac{1}{a+\frac{1}{b}}-\frac{4}{b\left ( abc+a+c \right )} \right )^{-\frac{1}{2}}=\left ( \frac{4}{a+\frac{c}{bc+1}}:\frac{b}{ab+1}-\frac{4}{b\left ( abc+a+c \right )} \right )^{-\frac{1}{2}}=\left ( \frac{4bc+4}{abc+a+c}\cdot \frac{ab+1}{b}-\frac{4}{b\left ( abc+a+c \right )} \right )^{-\frac{1}{2}}=\left ( \frac{4ab^{2}c+4bc+4ab+4}{b\left ( abc+a+c \right )} -\frac{4}{b\left ( abc+a+c \right )}\right )^{-\frac{1}{2}}=\left ( \frac{4ab^{2}c+4bc+4ab+4-4}{b\left ( abc+a+c \right )} \right )^{-\frac{1}{2}}=4^{-\frac{1}{2}}=\frac{1}{2}\)

Ответ: \(\frac{1}{2}\)

Упростить выражение \(\left ( \frac{2}{\sqrt{3}-1}+\frac{3}{\sqrt{3}-2}+\frac{15}{3-\sqrt{3}} \right )\left ( \sqrt{3}+5 \right )^{-1}\)

Решение №16948: \(\left ( \frac{2}{\sqrt{3}-1}+\frac{3}{\sqrt{3}-2}+\frac{15}{3-\sqrt{3}} \right )\left ( \sqrt{3}+5 \right )^{-1}=\left ( \frac{2\left ( \sqrt{3}+1 \right )}{2}+\frac{3\left ( \sqrt{3}+2 \right )}{-1}+\frac{15\left ( 3+\sqrt{3} \right )}{6} \right )\cdot \frac{1}{\sqrt{3}+5}=\frac{-4\sqrt{3}-10+15+5\sqrt{3}}{2}\cdot \frac{1}{\sqrt{3}+5}=\frac{\sqrt{3}+5}{2}\cdot \frac{1}{\sqrt{3}+5}=\frac{1}{2}\)

Ответ: \(\frac{1}{2}\)

Проверить справедливость равенств \(\frac{\sqrt{\sqrt[4]{8}-\sqrt{\sqrt{2}+1}}}{\sqrt{\sqrt[4]{8}+\sqrt{\sqrt{2}-1}}-\sqrt{\sqrt[4]{8}-\sqrt{\sqrt{2}-1}}}\)

Решение №16949: \(\frac{\sqrt{\sqrt[4]{8}-\sqrt{\sqrt{2}+1}}}{\sqrt{\sqrt[4]{8}+\sqrt{\sqrt{2}-1}}-\sqrt{\sqrt[4]{8}-\sqrt{\sqrt{2}-1}}}=\frac{1}{\sqrt{2}};\frac{\sqrt[4]{8}-\sqrt{\sqrt{2}+1}}{2\sqrt[4]{8}-2\sqrt{\sqrt{8}-\sqrt{2}+1}}=\frac{1}{2};\frac{\sqrt[4]{8}-\sqrt{\sqrt{2}+1}}{2\left ( \sqrt[4]{8}-\sqrt{\sqrt{2}+1}\right )}=\frac{1}{2};\frac{1}{2}=\frac{1}{2}\)

Ответ: \(\frac{1}{2}=\frac{1}{2}\\)

Упростить выражение и вычислить \(\frac{\sqrt{x^{2}y^{-2}-xy^{-1}+\frac{1}{4}}\left ( xy^{-2}+y^{-\frac{3}{2}} \right )}{2x^{2}-y^{\frac{3}{2}}-xy+2xy^{\frac{1}{2}}}\)

Решение №16950: \(\frac{\sqrt{x^{2}y^{-2}-xy^{-1}+\frac{1}{4}}\left ( xy^{-2}+y^{-\frac{3}{2}} \right )}{2x^{2}-y^{\frac{3}{2}}-xy+2xy^{\frac{1}{2}}}=\frac{\frac{\sqrt{\left ( 2x-y \right )^{2}}}{2y}\cdot \frac{x+\sqrt{y}}{y^{2}}}{\left ( x+\sqrt{y} \right )\left ( 2x-y \right )}=\frac{\left | 2x-y \right |}{2y^{3}\left ( 2x-y \right )}=-\frac{1}{2y^{3}};\frac{1}{2y^{3}}\)

Ответ: \(-\frac{1}{2y^{3}};\frac{1}{2y^{3}}\)

Упростить выражение \(\frac{\sqrt{c-d}}{c^{2}\sqrt{2c}}\left ( \sqrt{\frac{c-d}{c+d}}+\sqrt{\frac{c^{2}+cd}{c^{2}-cd}} \right )\)

Решение №16951: \(\frac{\sqrt{c-d}}{c^{2}\sqrt{2c}}\left ( \sqrt{\frac{c-d}{c+d}}+\sqrt{\frac{c^{2}+cd}{c^{2}-cd}} \right )=\frac{\sqrt{c-d}}{c^{2}\sqrt{2c}}\left (\sqrt{\frac{c-d}{c+d}}+\sqrt{\frac{c\left ( c+d \right )}{c\left ( c-d \right )}} \right )=\frac{\sqrt{c-d}}{c^{2}\sqrt{2c}}\left ( \sqrt{\frac{c-d}{c+d}}+\frac{\sqrt{c+d}}{\sqrt{c-d}} \right )=\frac{2c}{c^{2}\sqrt{2c}\sqrt{c+d}}=\frac{\sqrt{2}}{c\sqrt{c}\sqrt{c+d}}=\frac{\sqrt{2}}{c\sqrt{c^{2}+cd}}=\frac{\sqrt{2}}{2\sqrt{4+2\cdot \frac{1}{4}}}=\frac{1}{\sqrt{9}}=\frac{1}{3}\)

Ответ: \(\frac{1}{3}\)

Упростить выражение \(\frac{a^{-1}-b^{-1}}{a^{-3}+b^{-3}}:\frac{a^{2}b^{2}}{\left ( a+b \right )^{2}-3ab}\cdot \left ( \frac{a^{2}-b^{2}}{ab} \right )^{-1}\)

Решение №16952: \(\frac{a^{-1}-b^{-1}}{a^{-3}+b^{-3}}:\frac{a^{2}b^{2}}{\left ( a+b \right )^{2}-3ab}\cdot \left ( \frac{a^{2}-b^{2}}{ab} \right )^{-1}=\frac{\frac{1}{a}-\frac{1}{b}}{\frac{1}{a^{3}}-\frac{1}{b^{3}}}:\frac{a^{2}b^{2}}{a^{2}+2ab+b^{2}-3ab}\cdot \frac{ab}{a^{2}-b^{2}}=-\frac{ab}{\left ( a+b \right )^{2}}=-\frac{\left ( 1-\sqrt{2} \right )\left ( 1+\sqrt{2} \right )}{\left ( 1-\sqrt{2}+1+\sqrt{2} \right )^{2}}=\frac{1}{4}\)

Ответ: \(\frac{1}{4}\)

Сделать указанную подстановку и упростить результат \(\frac{x\left ( x+1 \right )\left ( x+2 \right )\left ( x+3 \right )}{\left ( x-1 \right )\left ( x+4 \right )}; x=\frac{\sqrt{5}-3}{2};\)

Решение №16953: \(\frac{x\left ( x+1 \right )\left ( x+2 \right )\left ( x+3 \right )}{\left ( x-1 \right )\left ( x+4 \right )}; x=\frac{\sqrt{5}-3}{2};=\frac{\frac{\sqrt{5}-3}{2}\left (\frac{\sqrt{5}-3}{2}+1 \right )\left (\frac{\sqrt{5}-3}{2}+2 \right )\left ( \frac{\sqrt{5}-3}{2}+3 \right )}{\left ( \frac{\sqrt{5}-3}{2}-1 \right )\left ( \frac{\sqrt{5}-3}{2}+4 \right )}=\frac{\left ( \left ( \frac{\sqrt{5}-3}{2} \right )^{2}+3\frac{\sqrt{5}-3}{2} \right )\left ( \left ( \frac{\sqrt{5}-3}{2} \right )^{2}+3\frac{\sqrt{5}-3}{2}+2 \right )}{\left ( \frac{\sqrt{5}-3}{2}-1 \right )^{2}+3\frac{\sqrt{5}-3}{2}-4}=\frac{\left ( \frac{7-3\sqrt{5}+3\sqrt{5}-9}{2} \right )^{2}+2\left (\frac{7-3\sqrt{5}+3\sqrt{5}-9}{2} \right )}{-1-4}=\frac{1}{5}\)

Ответ: \(\frac{1}{5}\)

Упростить выражение \(\frac{\left ( a^{\frac{1}{m}}-a^{\frac{1}{n}} \right )^{2}+4a^{\frac{m+n}{mn}}}{\left ( a^{\frac{2}{m}}-a^{\frac{2}{n}} \right )\left ( \sqrt[m]{a^{m+1}}+\sqrt[n]{a^{n+1}} \right )}\)

Решение №16954: \(\frac{\left ( a^{\frac{1}{m}}-a^{\frac{1}{n}} \right )^{2}+4a^{\frac{m+n}{mn}}}{\left ( a^{\frac{2}{m}}-a^{\frac{2}{n}} \right )\left ( \sqrt[m]{a^{m+1}}+\sqrt[n]{a^{n+1}} \right )}=\frac{a^{\frac{2}{m}}-2a^{\frac{1}{m}+\frac{1}{n}}+a^{\frac{2}{n}}+4a^{\frac{1}{m}+\frac{1}{n}}}{\left ( a^{\frac{1}{m}}-a\frac{1}{n} \right )\left ( a^{\frac{1}{m}}+a^{\frac{1}{n}} \right )\left ( a^{1+\frac{1}{m}}+a^{1+\frac{1}{n}} \right )}=\frac{\left ( a^{\frac{1}{m}}+a^{\frac{1}{n}} \right )^{2}}{\left ( a^{\frac{1}{m}}-a\frac{1}{n} \right )\left ( a^{\frac{1}{m}}+a^{\frac{1}{n}} \right )a\left ( a^{\frac{1}{m}}+a^{\frac{1}{n}} \right )}=\frac{1}{a\left ( a^{\frac{1}{m}}-a^{\frac{1}{n}} \right )}=\frac{1}{a\left ( \sqrt[m]{a}-\sqrt[n]{a} \right )}\)

Ответ: \(\frac{1}{a\left ( \sqrt[m]{a}-\sqrt[n]{a} \right )}\)

Упростить выражение и вычислить \(\frac{b^{-\frac{1}{6}}\sqrt{a^{3}b}\sqrt[3]{a^{3}b}-\sqrt{a^{3}b^{2}}\sqrt[3]{b^{2}}}{\left ( 2a^{2}-b^{2}-ab \right )\sqrt[6]{a^{9}b^{4}}}:\left ( \frac{3a^{3}}{2a^{2}-ab-b^{2}}-\frac{ab}{a-b} \right )\)

Решение №16955: \(\frac{b^{-\frac{1}{6}}\sqrt{a^{3}b}\sqrt[3]{a^{3}b}-\sqrt{a^{3}b^{2}}\sqrt[3]{b^{2}}}{\left ( 2a^{2}-b^{2}-ab \right )\sqrt[6]{a^{9}b^{4}}}:\left ( \frac{3a^{3}}{2a^{2}-ab-b^{2}}-\frac{ab}{a-b} \right )=\frac{\sqrt[6]{a^{15}b^{5}}-\sqrt[6]{a^{9}b^{10}}}{\left ( a-b \right )\left ( 2a+b \right )\sqrt[6]{a^{9}b^{4}}}:\frac{3a^{3}-ab\left ( 2a+b \right )}{\left ( a-b \right )\left ( 2a+b \right )}=\frac{1}{2a+b}\cdot \frac{\left ( a-b \right )2a+b}{3a^{3}-2a^{2}b-ab^{2}}=\frac{a-b}{\left ( a-b \right )\left ( 2a^{2}+a\left ( a+b \right ) \right )}=\frac{1}{3a^{2}+ab}=\frac{1}{a\left ( 3a+b \right )}\)

Ответ: \(\frac{1}{a\left ( 3a+b \right )}\)

Упростить выражение \(\frac{1}{2\left ( 1+\sqrt{a} \right )}+\frac{1}{2\left ( 1-\sqrt{a} \right )}-\frac{a^{2}+2}{1-a^{3}}\)

Решение №16956: \(\frac{1}{2\left ( 1+\sqrt{a} \right )}+\frac{1}{2\left ( 1-\sqrt{a} \right )}-\frac{a^{2}+2}{1-a^{3}}=\frac{1-\sqrt{a}+1+\sqrt{a}}{2\left ( 1+\sqrt{a} \right )\left ( 1-\sqrt{a} \right )}-\frac{a^{2}+2}{1-a^{3}}=\frac{2}{2\left ( 1-a \right )}-\frac{a^{2}+2}{\left ( 1-a \right )\left ( 1+a+a^{2} \right )}=\frac{1}{1-a}-\frac{a^{2}+2}{\left ( 1-a \right )\left ( 1+a+a^{2} \right )}=\frac{1+a+a^{2}-a^{2}-2}{\left ( 1-a \right )\left ( 1+a+a^{2} \right )}=\frac{-1}{a^{2}+a+1}\)

Ответ: \(\frac{-1}{a^{2}+a+1}\)

Упростить выражение и вычислить \(\frac{2\left | a+5 \right |-a+\frac{25}{a}}{3a^{2}+10a-25}\)

Решение №16957: \(\frac{2\left | a+5 \right |-a+\frac{25}{a}}{3a^{2}+10a-25}=\frac{2a\left | a+5 \right |-a^{2}+25}{a\left ( 3a^{2}+10a-25 \right )}=\frac{3\left ( 3a^{2}+10a-25 \right )}{a\left ( 3a^{2}+10a-25 \right )};\frac{a^{2}+10a+25}{a\left ( a+5 \right )\left ( 3a-5 \right )}=-\frac{1}{a};\frac{a+5}{a\left ( 3a-5 \right )}\)

Ответ: \(-\frac{1}{a};\frac{a+5}{a\left ( 3a-5 \right )}\)

Упростить выражение \(\left ( \frac{\sqrt[4]{a^{3}}-1}{\sqrt[4]{a}-1}+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \frac{\sqrt[4]{a^{3}}+1}{\sqrt[4]{a}+1}-\sqrt{a} \right )\cdot \left ( a-\sqrt{a^{3}} \right )^{-1}\)

Решение №16958: \(\left ( \frac{\sqrt[4]{a^{3}}-1}{\sqrt[4]{a}-1}+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \frac{\sqrt[4]{a^{3}}+1}{\sqrt[4]{a}+1}-\sqrt{a} \right )\cdot \left ( a-\sqrt{a^{3}} \right )^{-1}=\left ( \frac{\left ( \sqrt[4]{a}-1 \right )\left ( \sqrt[4]{a^{2}}+\sqrt[4]{a}+1 \right )}{\sqrt[4]{a}-1}+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \frac{\left ( \sqrt[4]{a}+1 \right )\left ( \sqrt[4]{a^{2}}-\sqrt[4]{a}+1 \right )}{\sqrt[4]{a}+1}-\sqrt{a} \right )\cdot \frac{1}{a-\sqrt{a^{3}}}=\left ( \sqrt[4]{a^{2}}+\sqrt[4]{a}+1+\sqrt[4]{a} \right )^{\frac{1}{2}}\cdot \left ( \sqrt[4]{a^{2}}-\sqrt[4]{a}+1-\sqrt[4]{a^{2}} \right )\cdot \frac{1}{a-\sqrt{a^{3}}}=\frac{\left ( \sqrt[4]{a}+1 \right )\left ( 1-\sqrt[4]{a} \right )}{a\left ( 1-\sqrt[4]{a} \right )\left ( 1+\sqrt[4]{a} \right )}=\frac{1}{a}\)

Ответ: \(\frac{1}{a}\)

Упростить выражение и вычислить \(\frac{a^{2}-4-\left | a-2 \right |}{a^{3}+2a^{2}-5a-6}\)

Решение №16959: \(\frac{a^{2}-4-\left | a-2 \right |}{a^{3}+2a^{2}-5a-6}=\frac{\left ( a-2 \right )\left ( a+2 \right )-\left | a-2 \right |}{\left ( a-2 \right )\left ( a+3 \right )\left ( a+1 \right )}=\frac{a+3}{\left ( a+3 \right )\left ( a+1 \right )};\frac{a+1}{\left ( a+3 \right )\left ( a+1 \right )}=\frac{1}{a+1};\frac{1}{a+3}\)

Ответ: \(\frac{1}{a+1};\frac{1}{a+3}\)

Упростить выражение \(\frac{\left ( \sqrt{a}+\sqrt{b} \right )^{2}-4b}{\left ( a-b \right ):\left ( \sqrt{\frac{1}{b}}+3\sqrt{\frac{1}{a}} \right )}:\frac{a+9b+6\sqrt{ab}}{\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{a}}}\)

Решение №16960: \(\frac{\left ( \sqrt{a}+\sqrt{b} \right )^{2}-4b}{\left ( a-b \right ):\left ( \sqrt{\frac{1}{b}}+3\sqrt{\frac{1}{a}} \right )}:\frac{a+9b+6\sqrt{ab}}{\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{a}}}=\frac{\frac{a+2\sqrt{ab}+b-4b}{\left ( \sqrt{a}-\sqrt{b} \right )\left ( \sqrt{a}+\sqrt{b} \right ):\left ( \frac{1}{\sqrt{b}}+\frac{3}{\sqrt{a}} \right )}}{\frac{\left ( \sqrt{a}+3\sqrt{b} \right )^{2}}{\frac{\sqrt{a}+\sqrt{b}}{\sqrt{ab}}}}=\frac{a+2\sqrt{ab}-3b}{\left ( \sqrt{a}-\sqrt{b} \right )\left ( \sqrt{a}+\sqrt{b} \right ):\frac{\sqrt{a}+3\sqrt{b}}{\sqrt{ab}}}:\frac{\left ( \sqrt{a}+3\sqrt{b} \right )^{2}\sqrt{ab}}{\left ( \sqrt{a}+\sqrt{b} \right )}=\frac{a+2\sqrt{ab}-3b}{ab\left ( a-\sqrt{ab}+3\sqrt{ab-3b} \right )}=\frac{1}{ab}\)

Ответ: \(\frac{1}{ab}\)

Упростить выражение и вычислить \(\left ( \left ( \frac{a\sqrt[3]{b}}{b\sqrt{a^{3}}} \right )^{\frac{3}{2}}+\left ( \frac{\sqrt{a}}{a\sqrt[8]{b^{3}}} \right )^{2} \right ):\left ( a^{\frac{1}{4}}+b^{\frac{1}{4}} \right )\)

Решение №16961: \(\left ( \left ( \frac{a\sqrt[3]{b}}{b\sqrt{a^{3}}} \right )^{\frac{3}{2}}+\left ( \frac{\sqrt{a}}{a\sqrt[8]{b^{3}}} \right )^{2} \right ):\left ( a^{\frac{1}{4}}+b^{\frac{1}{4}} \right )=\left ( \frac{a^{\frac{3}{2}}\left ( \sqrt[3]{b} \right )^{\frac{3}{2}}}{b^{\frac{3}{2}}\left ( \sqrt{a^{3}} \right )^{\frac{3}{2}}}+\frac{\left ( \sqrt{a} \right )^{2}}{a^{2}\left ( \sqrt[8]{b^{3}} \right )^{2}} \right ):\left ( a^{\frac{1}{4}}+b^{\frac{1}{4}} \right )=\left ( \frac{a^{\frac{3}{2}}b^{\frac{1}{2}}}{a^{\frac{9}{4}}b^{\frac{3}{2}}}+\frac{a}{a^{2}b^{\frac{3}{4}}} \right )\frac{1}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}=\left ( \frac{1}{a^{\frac{3}{4}}b}+\frac{1}{ab^{\frac{3}{4}}} \right )\frac{1}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}=\frac{1}{ab}\)

Ответ: \(\frac{1}{ab}\)

Упростить выражение \(\left ( \left ( \sqrt{mn}-\frac{mn}{m+\sqrt{mn}} \right ):\frac{\sqrt[4]{mn}-\sqrt{n}}{m-n}-m\sqrt{n} \right )^{2}:\sqrt[3]{mn\sqrt{mn}}-\left ( \frac{m}{\sqrt{m^{4}-1}} \right )^{-2}\)

Решение №16962: \(\left ( \left ( \sqrt{mn}-\frac{mn}{m+\sqrt{mn}} \right ):\frac{\sqrt[4]{mn}-\sqrt{n}}{m-n}-m\sqrt{n} \right )^{2}:\sqrt[3]{mn\sqrt{mn}}-\left ( \frac{m}{\sqrt{m^{4}-1}} \right )^{-2}=\left ( \sqrt{mn}\left ( 1-\frac{\sqrt{mn}}{\left ( \sqrt{m}+\sqrt{} \right )\sqrt{m}} \right ):\frac{\sqrt[4]{n}\left ( \sqrt[4]{m}-\sqrt[4]{n} \right )}{\left ( \sqrt{m}-\sqrt{n} \right )\left ( \sqrt{m}+\sqrt{n} \right )} -m\sqrt{n}\right )^{2}:\sqrt{mn}-\frac{m^{4}-1}{m^{2}}=\left ( m\sqrt[4]{n}\left ( \sqrt[4]{m}+\sqrt[4]{n} \right )-m\sqrt{n} \right )^{2}\cdot \frac{1}{\sqrt{mn}}-\frac{m^{4}-1}{m^{2}}=m^{2}-\frac{m^{4}-1}{m^{2}}=\frac{m^{4}-m^{4}+1}{m^{2}}=\frac{1}{m^{2}}\)

Ответ: \(\frac{1}{m^{2}}\)

Упростить выражение и вычислить \(\(\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}\)

Решение №16963: \(\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}; 1)\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}=\frac{-m\left ( m-3 \right )}{-\left ( m^{2}-m-6 \right )m}=\frac{m-3}{\left ( m-3 \right )\left ( m+2 \right )}=\frac{1}{m+2} 2)\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}=\frac{-m\left ( m-3 \right )}{\left ( m^{2}-m-6 \right )m}=\frac{-m+3}{\left ( m-3 \right )\left ( m+2 \right )}=-\frac{1}{m+2} 3)\frac{m\left | m-3 \right |}{\left ( m^{2}-m-6 \right )\left | m \right |}=\frac{-m\left ( m-3 \right )}{-\left ( m^{2}-m-6 \right )m}=\frac{1}{m+2} \frac{1}{m+2};-\frac{1}{m+2}\)

Ответ: \(\frac{1}{m+2};-\frac{1}{m+2}\)

Упростить выражение и вычислить \(\frac{\sqrt{4x+4+x^{-1}}}{\sqrt{x}\left | 2x^{2}-x-1 \right |}\)

Решение №16964: \(\frac{\sqrt{4x+4+x^{-1}}}{\sqrt{x}\left | 2x^{2}-x-1 \right |}=\frac{\sqrt{4x+4+\frac{1}{x}}}{\sqrt{x}\left | \left ( x-1 \right )\left ( 2x+1 \right ) \right |}=\frac{\sqrt{\left ( 2x+1 \right )^{2}}}{x\left | \left ( x-1 \right )\left ( 2x+1 \right ) \right |}=\frac{2x+1}{x\left ( 2x+1 \right )\left | x-1 \right |}=\frac{1}{x\left | x-1 \right |}=-\frac{1}{x\left ( x-1 \right )};\frac{1}{x\left ( x-1 \right )}\)

Ответ: \(-\frac{1}{x\left ( x-1 \right )};\frac{1}{x\left ( x-1 \right )}\)

Упростить выражение и вычислить \(\frac{\left ( \sqrt{x}+\sqrt{2} \right )^{2}-\sqrt{2x}}{x^{2}+x-\sqrt{2x}+2}\)

Решение №16965: \(\frac{\left ( \sqrt{x}+\sqrt{2} \right )^{2}-\sqrt{2x}}{x^{2}+x-\sqrt{2x}+2}=\frac{x^{2}+2\sqrt{2x}+2-\sqrt{2x}}{\left ( x^{2}+x\sqrt{2x}+2x \right )-\left ( x\sqrt{2x}+\sqrt{2x}\sqrt{2x}+2\sqrt{2x} \right )+\left ( x+\sqrt{2x}+2 \right )}=\frac{x+\sqrt{2x}+2}{x\left ( x+\sqrt{2x}+2 \right )-\sqrt{2x}\left ( x+\sqrt{2x}+2 \right )+\left ( x+\sqrt{2x}+2 \right )}=\frac{x+\sqrt{2x}+2}{\left ( x+\sqrt{2x}+2 \right )\left ( x-\sqrt{2x}+1 \right )}=\frac{1}{x-\sqrt{2x}+1}\)

Ответ: \(\frac{1}{x-\sqrt{2x}+1}\)

Упростить выражение \(\left ( x\sqrt[3]{\frac{x-1}{\left ( x+1 \right )^{2}}}+\frac{x-1}{\sqrt[3]{\left ( x^{2}-1 \right )^{2}}} \right )^{-\frac{3}{5}}:\left ( x^{2}-1 \right )^{\frac{4}{5}}\)

Решение №16966: \(\left ( x\sqrt[3]{\frac{x-1}{\left ( x+1 \right )^{2}}}+\frac{x-1}{\sqrt[3]{\left ( x^{2}-1 \right )^{2}}} \right )^{-\frac{3}{5}}:\left ( x^{2}-1 \right )^{\frac{4}{5}}=\left ( \frac{x\sqrt[3]{x-1}}{\sqrt[3]{\left ( x+1 \right )^{2}}}+\frac{\sqrt[3]{x-1}}{\sqrt[3]{\left ( x+1 \right )^{2}}} \right )^{-\frac{3}{5}}\cdot \frac{1}{\sqrt[5]{\left ( x+1 \right )^{4}}}=\left ( \frac{\sqrt[3]{x-1}\left ( x+1 \right )}{\sqrt[3]{\left ( x+1 \right )^{2}}} \right )^{-\frac{3}{5}}\cdot \frac{1}{\sqrt[5]{\left ( x^{2}-1 \right )^{4}}}=\left ( \sqrt[3]{\left ( x-1 \right )\left ( x+1 \right )} \right )^{-\frac{3}{5}}\cdot \frac{1}{\sqrt[5]{\left ( x^{2}-1 \right )^{4}}}=\frac{1}{\sqrt[5]{\left ( x^{2}-1 \right )^{5}}}=\frac{1}{x^{2}-1}\)

Ответ: \(\frac{1}{x^{2}-1}\)

Упростить выражение и вычислить \(\frac{x^{3}-6x^{2}+11x-6}{\left ( x^{3}-4x^{2}+3x \right )\left | x-2 \right |}\)

Решение №16967: \(\frac{x^{3}-6x^{2}+11x-6}{\left ( x^{3}-4x^{2}+3x \right )\left | x-2 \right |}=\frac{\left ( x-3 \right )\left ( x-2 \right )\left ( x-1 \right )}{x\left ( x-3 \right )\left ( x-1 \right )\left | x-2 \right |}=\frac{x-2}{x\left | x-2 \right |}=-\frac{1}{x};\frac{1}{x}\)

Ответ: \(-\frac{1}{x};\frac{1}{x}\)

Упростить выражение и вычислить \(\left ( \frac{z-2}{6z+\left ( z-2 \right )^{2}}+\frac{\left ( z+4 \right )^{2}-12}{z^{3}-8}-\frac{1}{z-2} \right ):\frac{z^{3}+2z^{2}+2z+4}{z^{3}-2z^{2}+2z-4}\)

Решение №16968: \(\left ( \frac{z-2}{6z+\left ( z-2 \right )^{2}}+\frac{\left ( z+4 \right )^{2}-12}{z^{3}-8}-\frac{1}{z-2} \right ):\frac{z^{3}+2z^{2}+2z+4}{z^{3}-2z^{2}+2z-4}=\left ( \frac{z-2}{6z+z^{2}-4z+4}+\frac{z^{2}+8z+16-12}{\left ( z-2 \right )\left ( z^{2}+2z+4 \right )}-\frac{2}{z-2} \right ):\frac{\left ( z+2 \right )\left ( z^{2}+2 \right )}{\left ( z-2 \right )\left ( z^{2}+2 \right )}=\frac{z^{2}+2z+4}{\left ( z+2 \right )\left ( z^{2}+2 \right )}\cdot \frac{z-2}{z+2}=\frac{1}{z+2}\)

Ответ: \(\frac{1}{z+2}\)

Упростить выражение \(\frac{2\sqrt{1+\frac{1}{4}\left ( \sqrt{\frac{1}{t}} -\sqrt{t}\right )^{2}}}{\sqrt{1+\frac{1}{4}\left ( \sqrt{\frac{1}{t}} -\sqrt{t}\right )^{2}-\frac{1}{2}\left ( \sqrt{\frac{1}{t}} -\sqrt{t}\right )}}\)

Решение №16969: \(\frac{2\sqrt{1+\frac{1}{4}\left ( \sqrt{\frac{1}{t}} -\sqrt{t}\right )^{2}}}{\sqrt{1+\frac{1}{4}\left ( \sqrt{\frac{1}{t}} -\sqrt{t}\right )^{2}-\frac{1}{2}\left ( \sqrt{\frac{1}{t}} -\sqrt{t}\right )}}=\frac{2\sqrt{1+\frac{1}{4}\left ( \frac{1}{t}-2+t \right )}}{\sqrt{1+\frac{1}{4}\left ( \frac{1}{t}-2+t \right )-\frac{1}{2}\left ( \frac{1}{\sqrt{t}}-\sqrt{t} \right )}}=\frac{2\sqrt{1+\frac{1}{4}\cdot \frac{1-2t+t^{2}}{t}}}{\sqrt{1+\frac{1}{4}\cdot \frac{1-2t+t^{2}}{t}-\frac{1}{2}\cdot \frac{1-t}{\sqrt{t}}}}=\frac{\sqrt{\frac{1+2t+t^{2}}{t}}}{\frac{1}{2}\sqrt{\frac{1+2t+t^{2}}{t}}-\frac{1-t}{2\sqrt{t}}}=\frac{\frac{1+t}{\sqrt{t}}}{\frac{1+t}{2\sqrt{t}}-\frac{1-t}{2\sqrt{t}}}=\frac{\frac{1+t}{\sqrt{t}}}{\frac{1+t-1+t}{2\sqrt{t}}}=\frac{1+t}{t}\)

Ответ: \(\frac{1+t}{t}\)

При каком значении k многочлен \(x^{2}+2\left ( k-9 \right )x+\left ( k^{2}+3k+4 \right )\) можно представить в виде полного квадрата?

Решение №16970: \(x^{2}+2\left ( k-9 \right )x+\left ( k^{2}+3k+4 \right )=\left ( 2\left ( k-9 \right ) \right )^{2}-4\left ( k^{2}+3k+4 \right )=0;4\left ( k^{2}-18k+81-k^{2}-3k-4 \right )=0;-21k+77=0;k=\frac{77}{21}=\frac{11}{3}\)

Ответ: \(\frac{11}{3}\)

Упростить выражение и вычислить \(\left ( \frac{\left ( x+\sqrt[3]{2ax^{2}} \right )\left ( 2a+\sqrt[3]{4a^{2}x} \right )^{-1}-1}{\sqrt[3]{x}-\sqrt[3]{2a}}\left ( 2a \right )^{-\frac{1}{3}} \right )^{-6}\)

Решение №16971: \(\left ( \frac{\left ( x+\sqrt[3]{2ax^{2}} \right )\left ( 2a+\sqrt[3]{4a^{2}x} \right )^{-1}-1}{\sqrt[3]{x}-\sqrt[3]{2a}}\left ( 2a \right )^{-\frac{1}{3}} \right )^{-6}=\left ( \frac{\frac{x+2^{\frac{1}{3}}a^{\frac{1}{3}}x^{\frac{2}{3}}}{2a+2^{\frac{2}{3}}a^{\frac{2}{3}}x^{\frac{1}{3}}}-1}{x^{\frac{1}{3}}-2^{\frac{1}{3}}a^{\frac{1}{3}}}-\frac{1}{2^{\frac{1}{3}}a^{\frac{1}{3}}} \right )^{-6}=\left ( \frac{x^{\frac{2}{3}}-2^{\frac{2}{3}}a^{\frac{2}{3}}}{2^{\frac{2}{3}}a^{\frac{2}{3}}\left ( x^{\frac{1}{3}-2^{\frac{1}{3}}a^{\frac{1}{3}}} \right )}-\frac{1}{2^{\frac{1}{3}}a^{\frac{1}{3}}} \right )^{-6}=\left ( \frac{x^{\frac{1}{3}}+2^{\frac{1}{3}}a^{\frac{1}{3}}-2^{\frac{1}{3}}a^{\frac{1}{3}}}{2^{\frac{2}{3}}a^{\frac{2}{3}}} \right )^{-6}=\left ( \frac{x^{\frac{1}{3}}}{2^{\frac{2}{3}}a^{\frac{2}{3}}} \right )^{-6}=\frac{16a^{4}}{x^{2}}\)

Ответ: \(\frac{16a^{4}}{x^{2}}\)

Упростить выражение \(\left ( \frac{1+\sqrt{x}}{\sqrt{1+x}}-\frac{\sqrt{1+x}}{1+\sqrt{x}} \right )^{2}-\left ( \frac{1-\sqrt{x}}{\sqrt{1+x}}-\frac{\sqrt{1+x}}{1-\sqrt{x}} \right )^{2}\)

Решение №16972: \(\left ( \frac{1+\sqrt{x}}{\sqrt{1+x}}-\frac{\sqrt{1+x}}{1+\sqrt{x}} \right )^{2}-\left ( \frac{1-\sqrt{x}}{\sqrt{1+x}}-\frac{\sqrt{1+x}}{1-\sqrt{x}} \right )^{2}=\left ( \frac{1+2\sqrt{x}+x-1-x}{\sqrt{1+x}\left ( 1+\sqrt{x} \right )} \right )^{2}-\left ( \frac{1-2\sqrt{x}+x-1-x}{\sqrt{1+x}\left ( 1-\sqrt{x} \right )} \right )^{2}=\left ( \frac{2\sqrt{x}}{\sqrt{1+x}\left ( 1+\sqrt{x} \right )} \right )^{2}-\left ( \frac{-2\sqrt{x}}{\sqrt{1+x}\left ( 1-\sqrt{x} \right )} \right )^{2}=\frac{4x}{\left (\sqrt{1+x}\left ( 1+\sqrt{x} \right )\right )^{2}}-\frac{4x}{\left (\sqrt{1+x}\left ( 1-\sqrt{x} \right )\right )^{2}}=\frac{-16x}{\left ( 1+x \right )\left ( 1-x \right )\left ( 1-x \right )}=\frac{16x\sqrt{x}}{\left ( 1-x^{2} \right )\left ( x-1 \right )}\)

Ответ: \(\frac{16x\sqrt{x}}{\left ( 1-x^{2} \right )\left ( x-1 \right )}\)

Упростить выражение и вычислить \(\left ( \sqrt[3]{\frac{x+1}{x-1}}+\sqrt[3]{\frac{x-1}{x+1}}-2 \right )^{\frac{1}{2}}\)

Решение №16973: \(\left ( \sqrt[3]{\frac{x+1}{x-1}}+\sqrt[3]{\frac{x-1}{x+1}}-2 \right )^{\frac{1}{2}}=\sqrt{\sqrt[3]{\frac{x+1}{x-1}}+\frac{1}{\sqrt[3]{\frac{x+1}{x-1}}}-2}=\left | \sqrt[3]{\frac{x+1}{x-1}}-1 \right |\sqrt[6]{\frac{x-1}{x+1}}=\left | \sqrt[3]{\frac{2a^{3}}{2}}-1 \right |\sqrt[6]{\frac{2}{2a^{3}}}=\left | a-1 \right |\frac{1}{\sqrt{a}}=\frac{1-a}{\sqrt{a}};\frac{a-1}{\sqrt{a}}\)

Ответ: \(\frac{1-a}{\sqrt{a}};\frac{a-1}{\sqrt{a}}\)

Упростить выражение \(\left ( \frac{\sqrt{a}}{2}-\frac{1}{2\sqrt{a}} \right )^{2}\cdot \left ( \frac{\sqrt{a}-1}{\sqrt{a}+1}-\frac{\sqrt{a}+1}{\sqrt{a}-1}\right )\)

Решение №16974: \(\left ( \frac{\sqrt{a}}{2}-\frac{1}{2\sqrt{a}} \right )^{2}\cdot \left ( \frac{\sqrt{a}-1}{\sqrt{a}+1}-\frac{\sqrt{a}+1}{\sqrt{a}-1}\right )=\left ( \frac{\left ( \sqrt{a} \right )^{2}-1}{2\sqrt{a}} \right )\cdot \frac{\left ( \sqrt{a}-1 \right )^{2}-\left ( \sqrt{a}+1 \right )^{2}}{\left ( \sqrt{a}+1 \right )\left ( \sqrt{a}-1 \right )}=\frac{\left ( a-1 \right )^{2}}{4a}\cdot \frac{a-2\sqrt{a}+1-a-2\sqrt{a}-1}{a-1}=\frac{\left ( a-1 \right )^{2}\left ( -4\sqrt{a} \right )}{4a\left ( a-1 \right )}=-\frac{a-1}{\sqrt{a}}=\frac{1-a}{\sqrt{a}}\)

Ответ: \(\frac{1-a}{\sqrt{a}}\)

Упростить выражение и вычислить \(\frac{\left ( x+1 \right )^{-\frac{1}{2}}}{\left ( x-1 \right )^{-\frac{1}{2}}-\left ( x+1 \right )^{-\frac{1}{2}}}\)

Решение №16975: \(\frac{\left ( x+1 \right )^{-\frac{1}{2}}}{\left ( x-1 \right )^{-\frac{1}{2}}-\left ( x+1 \right )^{-\frac{1}{2}}}=\frac{1}{\sqrt{x+1}}\cdot \frac{\sqrt{x-1}\sqrt{x+1}}{\sqrt{x+1}-\sqrt{x-1}}=\frac{\sqrt{x-1}\left ( \sqrt{x+1}+\sqrt{x-1} \right )}{\left ( \sqrt{x+1} \right )^{2}-\left ( \sqrt{x-1} \right )^{2}}=\frac{\sqrt{x-1}\left ( \sqrt{x+1}+\sqrt{x-1} \right )}{x+1-x+1}=\frac{\sqrt{x-1}\left ( \sqrt{x+1}+\sqrt{x-1} \right )}{2}=\frac{\sqrt{\frac{a^{2}+1}{2a}-1}\left ( \sqrt{\frac{a^{2}+1}{2a}+1} +\sqrt{\frac{a^{2}+1}{2a}-1}\right )}{2}=\frac{\left | a-1 \right |\left ( a+1 \right )+\left ( a-1 \right )^{2}}{4a}=\frac{1-a}{2a};\frac{a-1}{2}\)

Ответ: \(\frac{1-a}{2a};\frac{a-1}{2}\)

Упростить выражение и вычислить \(\frac{b^{2}-3b-\left ( b-1 \right )\sqrt{b^{2}-4}+1}{b^{2}+3b-\left ( b-1 \right )\sqrt{b^{2}-4}+1}\sqrt{\frac{b+2}{b-2}}\)

Решение №16976: \(\frac{b^{2}-3b-\left ( b-1 \right )\sqrt{b^{2}-4}+1}{b^{2}+3b-\left ( b-1 \right )\sqrt{b^{2}-4}+1}\sqrt{\frac{b+2}{b-2}}=\frac{\left ( b^{2}-3b+b \right )-\left ( b-1 \right )\sqrt{\left ( b-2 \right )\left ( b+2 \right )}}{\left ( b^{2}-3b+b \right )-\left ( b-1 \right )\sqrt{\left ( b-2 \right )\left ( b+2 \right )}{\sqrt{\frac{b+2}{b-2}}=-\frac{b-1}{b+1}=\frac{1-b}{1+b}\)

Ответ: \(\frac{1-b}{1+b}\)

Упростить выражение \((\left ( \left ( \sqrt[4]{p}-\sqrt[4]{q} \right )^{-2} +\left ( \sqrt[4]{p} +\sqrt[4]{q}\right )^{-2}\right ):\frac{\sqrt{p}+\sqrt{q}}{p-q}\)

Решение №16977: \(\left ( \left ( \sqrt[4]{p}-\sqrt[4]{q} \right )^{-2} +\left ( \sqrt[4]{p} +\sqrt[4]{q}\right )^{-2}\right ):\frac{\sqrt{p}+\sqrt{q}}{p-q}=\left ( \frac{1}{\left ( \sqrt[4]{p}-\sqrt[4]{q} \right )^{2}} +\frac{1}{\left ( \sqrt[4]{p}+\sqrt[4]{q} \right )^{2}}\right )\cdot \frac{p-q}{\sqrt{p}+\sqrt{q}}=\frac{\left ( \sqrt[4]{p}+\sqrt[4]{q} \right )^{2}+\left ( \sqrt[4]{p}-\sqrt[4]{q} \right )^{2}}{\sqrt{p}-\sqrt{q}}\cdot \frac{\left ( \sqrt{p}-\sqrt{q} \right )\left ( \sqrt{p}+\sqrt{q} \right )}{\sqrt{p}+\sqrt{q}}=\frac{\sqrt{p}+2\sqrt[4]{pq}+\sqrt{q}+\sqrt{p}-2\sqrt[4]{pq}+\sqrt{q}}{\sqrt{p}-\sqrt{q}}=\frac{2\left ( \sqrt{p}+\sqrt{q} \right )}{\sqrt{p}-\sqrt{q}}=\frac{2\left ( \sqrt{p}+\sqrt{q} \right )\left ( \sqrt{p}+\sqrt{q} \right )}{\left ( \sqrt{p}-\sqrt{q} \right )\left ( \sqrt{p}-\sqrt{q} \right )}=\frac{2\left ( \sqrt{p}+\sqrt{q} \right )^{2}}{p-q}\)

Ответ: \(\frac{2\left ( \sqrt{p}+\sqrt{q} \right )^{2}}{p-q}\)

Доказать, что если a+b=1, то \frac{a}{b^{3}-1}-\frac{b}{a^{3}-1}=\frac{2\left ( b-a \right )}{a^{2}b^{2}+3}

Решение №16978: \(\frac{a}{b^{3}-1}-\frac{b}{a^{3}-1}=\frac{\left ( a^{4}-b^{4} \right )-\left ( a-b \right )}{a^{3}b^{3}-\left ( a^{3}+b^{3} \right )+1}=\frac{\left ( a-b \right )\left ( 1-2ab-1 \right )}{a^{3}b^{3}-\left ( a^{3}+b^{3} \right )+1}=\frac{\left ( a-b \right )\left ( -2ab \right )}{a^{2}b^{3}+3ab}=\frac{2\left ( b-a \right )}{a^{2}b^{2}+3}\)

Ответ: \(\frac{2\left ( b-a \right )}{a^{2}b^{2}+3}\)

Упростить выражение \(\frac{\left ( \sqrt[3]{\left (r^{2}+4}\right )\cdot \sqrt{1+\frac{4}{r^{2}}}-\sqrt[3]{\left ( r^{2}+4 \right )\sqrt{1-\frac{4}{r^{2}}}} \right )^{2}}{r^{2}-\sqrt{r^{4}-16}}\)

Решение №16979: \(\frac{\left ( \sqrt[3]{\left (r^{2}+4}\right )\cdot \sqrt{1+\frac{4}{r^{2}}}-\sqrt[3]{\left ( r^{2}+4 \right )\sqrt{1-\frac{4}{r^{2}}}} \right )^{2}}{r^{2}-\sqrt{r^{4}-16}}=\frac{2\left ( r^{2}-\sqrt[3]{\left ( r^{4}-16 \right )^{\frac{3}{2}}} \right )}{\sqrt[3]{r^{2}}\left ( r^{2}-\sqrt{r^{4}-16} \right )}=\frac{2}{\sqrt[3]{r^{2}}}=\frac{2\sqrt[3]{r}}{r}\)

Ответ: \(\frac{2\sqrt[3]{r}}{r}\)

Упростить выражение \(\sqrt{\frac{2a}{\left ( 1+a \right )\sqrt[3]{1+a}}}\cdot \sqrt[3]{\frac{4+\frac{8}{a}+\frac{4}{a^{2}}}{\sqrt{2}}}\)

Решение №16980: \(\sqrt{\frac{2a}{\left ( 1+a \right )\sqrt[3]{1+a}}}\cdot \sqrt[3]{\frac{4+\frac{8}{a}+\frac{4}{a^{2}}}{\sqrt{2}}}=\sqrt[6]{\left ( \frac{2a}{\left ( 1+a \right )\sqrt[3]{1+a}} \right )^{3}}\cdot \sqrt[6]{\left ( \frac{\frac{4+\frac{8}{a}+\frac{4}{a^{2}}}{\sqrt{2}}} \right )}=\sqrt[6]{\frac{8a^{3}}{\left ( 1+a \right )^{4}}\cdot \sqrt[6]{\frac{8\left ( 1+a \right )^{4}}{a^{4}}}}=\sqrt[6]{\frac{64}{a}}=\frac{2\sqrt[6]{a^{5}}}{a}\)

Ответ: \(\frac{2\sqrt[6]{a^{5}}}{a}\)

Вычислить значение выражения \frac{z^{3}}{3}-z, z=\sqrt[3]{\sqrt{3}+\sqrt{2}}+\sqrt[3]{\sqrt{3}-\sqrt{2}}; x^{3}+3x, x=\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}

Решение №16981: \(\frac{z^{3}}{3}-z=\frac{\sqrt[3]{\sqrt{3}+\sqrt{2}}+\sqrt[3]{\sqrt{3}-\sqrt{2}}}{3}-\left ( \sqrt[3]{\sqrt{3}+\sqrt{2}}+\sqrt[3]{\sqrt{3}-\sqrt{2}} \right )=\frac{2\sqrt{3}+3\sqrt[3]{\left ( 3-2 \right )\left ( \sqrt{3}+\sqrt{2} \right )}-3\sqrt[3]{\left ( 3-2 \right )\left ( \sqrt{3}-\sqrt{2} \right )}-3\sqrt[3]{\sqrt{3}+\sqrt{2}}-3\sqrt[3]{\sqrt{3}-\sqrt{2}}}{3}=\frac{2\sqrt{3}+3\sqrt[3]{\sqrt{3}+\sqrt{2}}+3\sqrt[3]{\sqrt{3}-\sqrt{2}}-3\sqrt[3]{\sqrt{3}+\sqrt{2}}-3\sqrt[3]{\sqrt{3}-\sqrt{2}}}{3}=\frac{2\sqrt{3}}{3}; x^{3}+3x=\left ( \sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2} \right )^{3}+3\left ( \sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2} \right )=\sqrt{5}+2-3\sqrt[3]{\left ( \sqrt{5}+2 \right )^{2}\left ( \sqrt{5}-2 \right )}+3\sqrt[3]{\left ( \sqrt{5}+2 \right )\left ( \sqrt{5}-2 \right )^{2}}-\sqrt{5}+2+3\sqrt[3]{\sqrt{5}+2}-3\sqrt[3]{\sqrt{5}-2}=4-3\sqrt[3]{\sqrt{5}+2}+3\sqrt[3]{\sqrt{5}-2}+3\sqrt[3]{\sqrt{5}+2}-3\sqrt[3]{\sqrt{5}-2}=4\)

Ответ: \(\frac{2\sqrt{3}}{3}; 4\)

Освободиться от иррациональности в дроби \(\frac{6}{\sqrt{2}+\sqrt{3}+\sqrt{5}}\)

Решение №16982: \(\frac{6}{\sqrt{2}+\sqrt{3}+\sqrt{5}}=\frac{6\left ( \sqrt{2}+\sqrt{3}-\sqrt{5} \right )}{\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )\left ( \sqrt{2}+\sqrt{3}-\sqrt{5} \right )}=\frac{6\left ( \sqrt{2}+\sqrt{3}-\sqrt{5} \right )}{2+3-5+2\sqrt{2*3}}=\frac{\sqrt{4*3}+\sqrt{9*2}-\sqrt{30}}{2}=\frac{2\sqrt{3}+3\sqrt{2}-\sqrt{30}}{2}\)

Ответ: \(\frac{2\sqrt{3}+3\sqrt{2}-\sqrt{30}}{2}\)

Упростить выражение \(\left ( \left ( 1-p^{2} \right )^{-\frac{1}{2}} -\left ( 1+p^{2} \right )^{-\frac{1}{2}} \right )^{2}+2\left ( 1-p^{4} \right )^{-\frac{1}{2}}\)

Решение №16983: \(\left ( \left ( 1-p^{2} \right )^{-\frac{1}{2}} -\left ( 1+p^{2} \right )^{-\frac{1}{2}} \right )^{2}+2\left ( 1-p^{4} \right )^{-\frac{1}{2}}=\left ( \frac{1}{\sqrt{1-p^{2}}}-\frac{1}{\sqrt{1+p^{2}}} \right )^{2}+\frac{2}{\sqrt{1-p^{4}}}=\left ( \frac{\sqrt{1+p^{2}}-\sqrt{1-p^{2}}}{\sqrt{1-p^{4}}} \right )^{2}+\frac{2}{\sqrt{1-p^{4}}}=\frac{1+p^{2}-2\sqrt{1-p^{4}}+1-p^{2}}{1-p^{4}}+\frac{2}{\sqrt{1-p^{4}}}=\frac{2-2\sqrt{1-p^{4}}}{1-p^{4}}+\frac{2}{\sqrt{1-p^{4}}}=\frac{2-2\sqrt{1-p^{4}}+2\sqrt{1-p^{4}}}{1-p^{4}}=\frac{2}{1-p^{4}}\)

Ответ: \(\frac{2}{1-p^{4}}\)

Упростить выражение \(\frac{x\left ( x^{2}-a^{2} \right )^{-\frac{1}{2}}+1}{a\left ( x-a \right )^{-\frac{1}{2}}+\left ( x+a \right )^{\frac{1}{2}}}:\frac{a^{2}\sqrt{x+a}}{x-\left ( x^{2}-a^{2} \right )^{\frac{1}{2}}}+\frac{1}{x^{2}-ax}\)

Решение №16984: \(\frac{x\left ( x^{2}-a^{2} \right )^{-\frac{1}{2}}+1}{a\left ( x-a \right )^{-\frac{1}{2}}+\left ( x+a \right )^{\frac{1}{2}}}:\frac{a^{2}\sqrt{x+a}}{x-\left ( x^{2}-a^{2} \right )^{\frac{1}{2}}}+\frac{1}{x^{2}-ax}=\frac{\frac{x}{\sqrt{x^{2}-a^{2}}+1}}{\frac{a}{\sqrt{x-a}}+\sqrt{x-a}}:\frac{a^{2}\sqrt{x+a}}{x-\sqrt{x^{2}-a^{2}}}+\frac{1}{x\left ( x-a \right )}=\frac{\left ( x+\sqrt{x^{2}-a^{2}}\sqrt{x-a} \right )}{\sqrt{\left ( x+a \right )\left ( x-a \right )}\left ( a+x-a \right )}\cdot \frac{x-\sqrt{x^{2}-a^{2}}}{a^{2}\sqrt{x+a}}+\frac{1}{x\left ( x-a \right )}=\frac{x^{2}-x^{2}+a^{2}}{a^{2}x\left ( x+a \right )}+\frac{1}{\left ( x-a \right )}=\frac{1}{\left ( x+a \right )}+\frac{1}{\left ( x-a \right )}=\frac{2}{x^{2}-a^{2}}\)

Ответ: \(\frac{2}{x^{2}-a^{2}}\)

Упростить выражение \(\frac{4a^{2}-b^{2}}{a^{6}-8b^{6}}\sqrt{a^{2}-2b\sqrt{a^{2}-b^{2}}}\cdot \frac{a^{4}+2a^{2}b^{2}+4b^{4}}{4a^{2}+4ab+b^{2}}\cdot \sqrt{a^{2}+2b\sqrt{a^{2}-b^{2}}}\)

Решение №16985: \(\frac{4a^{2}-b^{2}}{a^{6}-8b^{6}}\sqrt{a^{2}-2b\sqrt{a^{2}-b^{2}}}\cdot \frac{a^{4}+2a^{2}b^{2}+4b^{4}}{4a^{2}+4ab+b^{2}}\cdot \sqrt{a^{2}+2b\sqrt{a^{2}-b^{2}}}=\frac{\left ( 2a-b \right )\left ( 2a+b \right )}{\left ( a^{2} \right )^{3}-\left ( 2b^{2} \right )^{3}}\cdot \frac{a^{4}+2a^{2}b^{2}+4b^{4}}{\left ( 2a+b \right )^{2}}\cdot \sqrt{\left (a^{2}-2b\sqrt{a^{2}-b^{2}} \right )\left (a^{2}+2b\sqrt{a^{2}-b^{2}}\right )}=\frac{\left ( 2a-b \right )\left ( a^{4}+2a^{2}b^{2}+4b^{4} \right )}{\left ( a^{2}-2b^{2} \right )\left ( a^{4}+2a^{2}b^{2}+4b^{4} \right )\left ( 2a+b \right )}\cdot \sqrt{a^{4}-4b^{2}\left ( a^{2}-b^{2} \right )}=\frac{\left ( 2a-b \right )\left ( a^{2}-2b^{2} \right )}{\left ( a^{2}-2b^{2} \right )\left ( 2a+b \right )}=\frac{2a-b}{2a+b}=\frac{2\cdot \frac{4}{3}-0.25}{2\cdot \frac{4}{3}+0.25}=\frac{7.25}{8.75}=\frac{29}{35}\)

Ответ: \(\frac{29}{35}\)

Сделать указанную подстановку и упростить результат \(\frac{\left ( z-1 \right )\left ( z+2 \right )\left ( z-3 \right )\left ( z+4 \right )}{23}; x=\frac{\sqrt{3}-1}{2};\)

Решение №16986: \(\frac{\left ( z-1 \right )\left ( z+2 \right )\left ( z-3 \right )\left ( z+4 \right )}{23}; x=\frac{\sqrt{3}-1}{2};=\frac{\left (\frac{\sqrt{3}-1}{2}-1 \right )\left ( \frac{\sqrt{3}-1}{2}+2 \right )\left ( \frac{\sqrt{3}-1}{2}-3 \right )\left ( \frac{\sqrt{3}-1}{2}+4 \right )}{23}=\frac{\left ( \left ( \frac{\sqrt{3}-1}{2} \right )^{2}+\frac{\sqrt{3}-1}{2}-2 \right )\left ( \left ( \frac{\sqrt{3}-1}{2} \right )^{2}+\frac{\sqrt{3}-1}{2}-12 \right )}{23}=\frac{\left ( \frac{1}{2} \right )^{2}-14\frac{1}{2}+24}{23}=\frac{\frac{1}{4}-7+24}{23}=\frac{3}{4}\)

Ответ: \(\frac{3}{4}\)