Решение №16086: \(\left ( \frac{a-\sqrt{a^{2}-b^{2}}}{a+\sqrt{a^{2}-b^{2}}}-\frac{a+\sqrt{a^{2}-b^{2}}}{a-\sqrt{a^{2}-b^{2}}} \right ):\frac{4\sqrt{a^{4}-a^{2}b^{2}}}{\left ( 5b \right )^{2}}=\frac{\left ( a-\sqrt{a^{2}-b^{2}} \right )^{2}-\left ( a+\sqrt{a^{2}-b^{2}} \right )^{2}}{\left ( a+\sqrt{a^{2}-b^{2}} \right )\left ( a-\sqrt{a^{2}-b^{2}} \right )}\cdot \frac{25b^{2}}{4\sqrt{a^{2}\left ( a^{2}-b^{2} \right )}}=\frac{a^{2}-2a\sqrt{a^{2}-b^{2}}+a^{2}-b^{2}-a^{2}-2a\sqrt{a^{2}-b^{2}}-a^{2}+b^{2}}{a^{2}-a^{2}+b^{2}}\cdot \frac{25b^{2}}{4\left | a \right |\sqrt{a^{2}-b^{2}}}=\frac{4a\sqrt{a^{2}-b^{2}}}{b^{2}}\cdot \frac{25b^{2}}{4\left | a \right |\sqrt{a^{2}-b^{2}}}=-\frac{25}{\left | a \right |}=-25,25\)

Ответ: -25.25

Решение №16087: \(t\cdot \frac{1+\frac{2}{\sqrt{t+4}}}{2-\sqrt{t+4}}+\sqrt{t+4}+\frac{4}{\sqrt{t+4}}=t\cdot \frac{\frac{\sqrt{t+4}+2}{\sqrt{t+4}}}{2-\sqrt{t+4}}+\frac{\left ( \sqrt{t+4} \right )^{2}+4}{\sqrt{t+4}}=t\cdot \frac{\sqrt{t+4}+2}{\sqrt{t+4}\left ( 2-\sqrt{t+4} \right )}+\frac{\left ( \sqrt{t+4} \right )^{2}+4}{\sqrt{t+4}}=\frac{t\left ( \sqrt{t+4}+2 \right )^{2}}{\sqrt{t+4}\left ( 4-t-4 \right )}+\frac{\left ( \sqrt{t+4} \right )^{2}+4}{\sqrt{t+4}}=\frac{-\left ( t+4+4\sqrt{t+4}+4 \right )}{\sqrt{t+4}}+\frac{t+4+4}{\sqrt{t+4}}=-\frac{4\sqrt{t+4}}{\sqrt{t+4}}=-4\)

Ответ: -4

Решение №16088: \(\frac{9b^{\frac{4}{3}}-\frac{a^{\frac{3}{2}}}{b^{2}}}{\sqrt{a^{\frac{3}{2}}+6a^{\frac{3}{4}}b^{-\frac{1}{3}}+9b^{\frac{4}{3}}}}\cdot \frac{b^{2}}{a^{\frac{3}{4}}-3b^{\frac{5}{3}}}=\frac{9b^{\frac{10}{3}}-a^{\frac{3}{2}}}{\sqrt{\frac{a^{\frac{3}{2}}+6a^{\frac{3}{4}}b^{\frac{5}{3}}+9b^{\frac{10}{3}}}{b^{2}}}\left ( a^{\frac{3}{4}}-3b^{\frac{5}{3}} \right )}=\frac{-\left ( a^{\frac{3}{4}}-3b^{\frac{5}{3}} \right )\left ( a^{\frac{3}{4}}+3b^{\frac{5}{3}} \right )}{\frac{\sqrt{\left ( a^{\frac{3}{4}}+3b^{\frac{5}{3}} \right )^{2}}}{b}\left ( a^{\frac{3}{4}}-3b^{\frac{5}{3}} \right )}=\frac{-\left ( a^{\frac{3}{4}} 3b^{\frac{5}{3}} \right )b}{a^{\frac{3}{4}}+3b^{\frac{5}{3}}}=-b=-4\)

Ответ: -4

Решение №16089: \(\sqrt{\frac{\sqrt{2}}{a}+\frac{a}{\sqrt{2}}+2}-\frac{a^{2}\sqrt[4]{2}-2\sqrt{a}}{a\sqrt{2a}-\sqrt[4]{8a^{4}}}=\sqrt{\frac{a^{2}+2a\sqrt{2}+\sqrt{2^{2}}}{a\sqrt{2}}}-\frac{a^{2}\cdot 2^{\frac{1}{4}}-2a^{\frac{1}{2}}}{a^{\frac{3}{2}}\cdot 2^{\frac{1}{2}}-2^{\frac{3}{4}}a}=\frac{a+\sqrt{2}}{a^{\frac{1}{2}}\cdot 2^{\frac{1}{4}}}-\frac{2^{\frac{1}{4}}a^{\frac{1}{2}}\left ( \left ( a^{\frac{1}{2}} \right )^{3}-\left ( 2^{\frac{1}{4}} \right )^{3} \right )}{2^{\frac{1}{2}}a\left ( a^{\frac{1}{2}}-2^{\frac{1}{4}} \right )}=\frac{a+2^{\frac{1}{2}}}{a^{\frac{1}{2}}\cdot 2^{\frac{1}{4}}}-\frac{a+2^{\frac{1}{4}}a^{\frac{1}{2}}+2^{\frac{1}{2}}}{a^{\frac{1}{2}}2^{\frac{1}{4}}}=\frac{-2^{\frac{1}{4}}a^{\frac{1}{2}}}{a^{\frac{1}{2}}2^{\frac{1}{4}}}=-1\)

Ответ: -1

Решение №16090: \(\frac{\sqrt{1-x^{2}}-1}{x}\cdot \left ( \frac{1-x}{\sqrt{1-x^{2}}+x-1}+\frac{\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}} \right )=\frac{\sqrt{1-x^{2}}-1}{x}\cdot \left ( \frac{\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}+\frac{\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}} \right )=\frac{\sqrt{1-x^{2}}-1}{x}\cdot \frac{\sqrt{1-x}+\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}}=\frac{\sqrt{1-x^{2}}-1}{x}\cdot \frac{1-x+2\sqrt{1-x^{2}}+1+x}{1+x-1+x}=\frac{1-x^{2}-1}{x^{2}}=\frac{-x^{2}}{x^{2}}=-1\)

Ответ: -1

Решение №16091: \(\frac{1}{b\left ( abc+a+c \right )}-\frac{1}{a+\frac{1}{b+\frac{1}{c}}}:\frac{1}{a+\frac{1}{b}}=\frac{1}{b\left ( abc+a+c \right )}-\frac{1}{a+\frac{c}{bc+1}}:\frac{b}{ab+1}=\frac{1}{b\left ( abc+a+c \right )}-\frac{bc+1}{abc+a+c}\cdot \frac{ab+1}{b}=\frac{1-ab^{2}c-ab-bc-1}{b\left ( abc+a+c \right )}=\frac{-b\left ( abc+a+c \right )}{b\left ( abc+a+c \right )}=-1\)

Ответ: -1

Решение №16092: \(\frac{\left ( x^{\frac{2}{3}}+2\sqrt[3]{xy}+4y^{\frac{2}{3}} \right )}{\left ( \sqrt[3]{x^{4}}-8y\sqrt[3]{x} \right ):\sqrt[3]{xy}}\cdot \left ( 2-\sqrt[3]{\frac{x}{y}} \right )=-\frac{y^{\frac{1}{3}}}{2y^{\frac{1}{3}}-x^{\frac{1}{3}}}\cdot \frac{2y^{\frac{1}{3}}-x^{\frac{1}{3}}}{y^{\frac{1}{3}}}=-1\)

Ответ: -1

Решение №16093: \(\left ( \frac{1+\sqrt{1-x}}{1-x+\sqrt{1-x}}+\frac{1-\sqrt{1-x}}{1+x-\sqrt{1-x}} \right )^{2}\cdot \frac{x^{2}-1}{2}-\sqrt{1-x^{2}}=\left ( \frac{1+\sqrt{1-x}}{\sqrt{1-x}\left ( \sqrt{1-x}+1 \right )}+\frac{1-\sqrt{1-x}}{\sqrt{1+x}\left ( \sqrt{1+x}-1 \right )} \right )\cdot \frac{x^{2}-1}{2}-\sqrt{1-x^{2}}=\frac{1+x-2\sqrt{1-x^{2}}+1-x}{1-x^{2}}\cdot \frac{x^{2}-1}{2}-\sqrt{1-x^{2}}=-1+\sqrt{1-x^{2}}-\sqrt{1-x^{2}}=-1\)

Ответ: -1

Решение №16094: \(\left ( \frac{\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}}+\frac{1-x}{\sqrt{1-x^{2}}-1+x} \right )\left ( \sqrt{\frac{1}{x^{2}}-1}-\frac{1}{x} \right )=\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}\cdot \frac{\sqrt{1-x^{2}}-1}{x}=\frac{\left ( \sqrt{1+x}+\sqrt{1-x} \right )\left ( \sqrt{1+x}+\sqrt{1-x} \right )}{\left ( \sqrt{1+x}-\sqrt{1-x} \right )\left ( \sqrt{1+x}+\sqrt{1-x} \right )}\cdot \frac{\sqrt{1-x^{2}}-1}{x}=\frac{1+x+2\sqrt{1-x^{2}}+1-x}{1+x-1+x}\cdot \frac{\sqrt{1-x^{2}}-1}{x}=\frac{\left ( \sqrt{1-x^{2}} \right )^{2}-1}{x^{2}}=\frac{1-x^{2}-1}{x^{2}}=-\frac{x^{2}}{x^{2}}=-1\)

Ответ: -1

Решение №16095: \(\frac{2x^{-\frac{1}{3}}}{x^{\frac{2}{3}}-3x^{-\frac{1}{3}}}-\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}-x^{\frac{2}{3}}}-\frac{x+1}{x^{2}-4x+3}=\frac{2x^{-\frac{1}{3}}}{x^{-\frac{1}{3}}-\left ( x-3 \right )}-\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}-\left ( x-1 \right )}-\frac{x+1}{\left ( x-1 \right )\left ( x-3 \right )}=\frac{2}{x-3}-\frac{1}{x-1}-\frac{x+1}{\left ( x-1 \right )\left ( x-3 \right )}=\frac{2x-2-x+3-x-1}{\left ( x-1 \right )\left ( x-3 \right )}=\frac{0}{\left ( x-1 \right )\left ( x-3 \right )}=0\)

Ответ: 0

Решение №16096: \(\frac{m^{\frac{4}{3}}-27m^{\frac{1}{3}}\cdot n}{m^{\frac{2}{3}}+3\sqrt[3]{mn}+9n^{\frac{2}{3}}}:\left ( 1-3\sqrt[3]{\frac{n}{m}} \right )-\sqrt[3]{m^{2}}=\frac{m^{\frac{1}{3}}\left ( m-27n \right )}{m^{\frac{2}{3}}+3m^{\frac{1}{3}}n^{\frac{1}{3}}+9n^{\frac{2}{3}}}:\frac{\sqrt[3]{m}-3\sqrt[3]{n}}{\sqrt[3]{m}}-m^{\frac{2}{3}}=m^{\frac{2}{3}}-m^{\frac{2}{3}}=0\)

Ответ: 0

Решение №16097: \(2\sqrt{40\sqrt{12}}+3\sqrt{5\sqrt{48}}-2\sqrt[4]{75}-4\sqrt{15\sqrt{27}}=2\sqrt{40\sqrt{4\cdot 3}}+3\sqrt{5\sqrt{16\cdot 3}}-2\sqrt[4]{25\cdot 3}-4\sqrt{15\sqrt{9\cdot 3}}=8\sqrt{5\sqrt{3}}+6\sqrt{5\sqrt{3}}-2\sqrt{5\sqrt{3}}-12\sqrt{5\sqrt{3}}=14\sqrt{5\sqrt{3}}-14\sqrt{5\sqrt{3}}=0\)

Ответ: 0

Решение №16098: \(5\sqrt[3]{6\sqrt{32}}-3\sqrt[3]{9\sqrt{162}}-11\sqrt[6]{18}+2\sqrt[3]{75\sqrt{50}}=5\sqrt[3]{6\sqrt{16*2}}-3\sqrt[3]{9\sqrt{81*2}}-11\sqrt[6]{9*2}+2\sqrt[3]{75\sqrt{25*2}}=5*2\sqrt[3]{3\sqrt{2}}-3*3\sqrt[3]{3\sqrt{2}}-11\sqrt[3]{3\sqrt{2}}+2*5\sqrt[3]{3\sqrt{2}}=10\sqrt[3]{3\sqrt{2}}-20\sqrt[3]{3\sqrt{2}}+10\sqrt[3]{3\sqrt{2}}=0\)

Ответ: 0

Решение №16099: \(\frac{x^{3}-a^{-\frac{2}{3}}b^{-1}\left ( a^{2}+b^{2} \right )x+b^{\frac{1}{2}}}{b^{\frac{3}{2}}x^{2}}; x=a^{\frac{2}{3}}b^{-\frac{1}{2}};=\frac{\left (a^{\frac{2}{3}}b^{-\frac{1}{2}} \right )^{3}-a^{\frac{2}{3}}b^{-1}\left ( a^{2}+b^{2} \right )a^{\frac{2}{3}}b^{-\frac{1}{2}}+b^{\frac{1}{2}}}{b^{\frac{3}{2}}\left ( a^{\frac{2}{3}}b^{-\frac{1}{2}} \right )^{2}}=\frac{\frac{a^{2}-a^{2}-b^{2}+b^{2}}{b^{\frac{3}{2}}}}{b^{\frac{1}{2}}a^{\frac{4}{3}}}=0\)

Ответ: 0

Решение №16100: \(\frac{1-b}{\sqrt{b}}\cdot x^{2}-2x+\sqrt{b}; x=\frac{\sqrt{b}}{1-\sqrt{b}};=\frac{1-b}{\sqrt{b}}\left ( \frac{\sqrt{b}}{1-\sqrt{b}} \right )^{2}-2\cdot \frac{\sqrt{b}}{1-\sqrt{b}}+\sqrt{b}=\frac{\left ( 1-\sqrt{b} \right )\left ( 1+\sqrt{b} \right )}{\sqrt{b}}\cdot \frac{b}{\left ( 1-\sqrt{b} \right )^{2}}-\frac{2\sqrt{b}}{1-\sqrt{b}}+\sqrt{b}=\frac{\left ( 1+\sqrt{b} \right )\sqrt{b}}{1-\sqrt{b}}-\frac{2\sqrt{b}}{1-\sqrt{b}}+\sqrt{b}=\frac{\sqrt{b}+b-2\sqrt{b}+\sqrt{b}-b}{1-\sqrt{b}}=0\)

Ответ: 0

Решение №16101: \(\left ( \sqrt[3]{\frac{8z^{3}+24z^{2}+18z}{2z-3}}-\sqrt[3]{\frac{8z^{2}-24z^{2}+18z}{2z+3}} \right )-\left ( \frac{1}{2}\sqrt[3]{\frac{2z}{27}}-\frac{1}{6z} \right )^{-1}=\sqrt[3]{\frac{2z\left ( 2z+3 \right )^{2}}{2z-3}}-\sqrt[3]{\frac{2z\left ( 2z-3 \right )^{2}}{2z+3}}-2\sqrt[3]{\frac{54z}{4z^{2}-9}}= \sqrt[3]{\frac{2z\left ( 2z+3 \right )^{2}}{2z-3}}-\sqrt[3]{\frac{2z\left ( 2z-3 \right )^{2}}{2z+3}}-\frac{2\sqrt[3]{54z}}{\sqrt[3]{\left ( 2z-3 \right )\left ( 2z+3 \right )}} =\frac{\sqrt[3]{2z}\left ( 2z+3-2z+3-6 \right )}{\sqrt[3]{4z^{2}-9}}=\frac{\sqrt[3]{2z}\cdot 0}{\sqrt[3]{4z^{2}-9}}=0\)

Ответ: 0

Решение №16102: \(\left ( \frac{1+1+x^{2}}{2x+x^{x}} +2-\frac{1-x+x^{2}}{2x-x^{2}}\right )^{-1}\cdot \left ( 5-2x^{2} \right )=\left ( \frac{1+1+x^{2}}{x\left ( 2+x \right )} +2-\frac{1-x+x^{2}}{x\left ( 2-x \right )}\right )^{-1}\cdot \left ( 5-2x^{2} \right )=\left ( \frac{2+2x+2x^{2}-x-x^{2}-x^{3}+8x-2x^{3}-2+2x-2x^{2}-x+x^{2}-x^{3}}{x\left ( 4-x^{2} \right )} \right )^{-1}\cdot \left ( 5-2x^{2} \right )=\left ( \frac{10x-4x^{3}}{x\left ( 4-x^{2} \right )} \right )^{-1}\cdot \left ( 5-2x^{2} \right )=\left ( \frac{2x\left ( 5-2x^{2} \right )}{x\left ( 4-x^{2} \right )} \right )^{-1}\cdot \left ( 5-2x^{2} \right )=\frac{4-x^{2}}{2}=\frac{4-\left ( \sqrt{3.92} \right )^{2}}{2}=\frac{0.08}{2}=0.04\)

Ответ: 0.04

Решение №16103: \(\frac{\frac{1}{a}-\frac{1}{b+c}}{\frac{1}{a}+\frac{1}{b+c}}\cdot \left ( 1+\frac{b^{2}+c^{2}-a^{2}}{2bc} \right ):\frac{a-b-c}{abc}=\frac{\left ( b+c-a \right )a\left ( b+c \right )}{a\left ( b+c \right )\left ( b+c+a \right )}\cdot \frac{\left ( b^{2}+2bc+c^{2} \right )-a^{2}}{2bc}\cdot \frac{abc}{a-b-c}=\frac{b+c-a}{b+c+a}\cdot \frac{\left ( b+c \right )^{2}-a^{2}}{2}\cdot \frac{ab}{a-b-c}=\frac{-\left ( a-b-c \right )\left ( b+c+a \right )\left ( b+c-a \right )a}{2\left ( b+c+a \right )\left ( a-b-c \right )}=\frac{-\left ( b+c-a \right )a}{2}=\frac{\left ( a-b-c \right )a}{2}=\frac{\left ( 0.02+11.05-1.07 \right )\cdot 0.02}{2}=0.1\)

Ответ: 0.1

Решение №16104: \(\left ( \frac{\left ( a+b \right )^{\frac{n}{4}}\cdot c^{\frac{1}{2}}}{a^{2-n}b^{-\frac{3}{4}}} \right )^{\frac{4}{3}}:\left ( \frac{b^{3}c^{4}}{\left ( a+b \right )^{2n}a^{16-8n}} \right )=\frac{bc^{\frac{2}{3}}}{a^{\frac{\left ( 8-4n \right )}{3}}\left ( a+b \right )^{\frac{n}{3}}}:\frac{b^{\frac{1}{2}}c^{\frac{2}{3}}}{\left ( a+b \right )^{\frac{n}{3}}\cdot a^{\left ( 8-\frac{4n}{3} \right )}}=\frac{bc^{\frac{2}{3}}\left ( a+b \right )^{\frac{n}{3}}\cdot a^{\left ( 8-\frac{4n}{3} \right )}{a^{\frac{\left ( 8-4n \right )}{3}}\left ( a+b \right )^{\frac{n}{3}}b^{\frac{1}{2}}c^{\frac{2}{3}}}=b^{\frac{1}{2}}=0.04^{\frac{1}{2}}=\sqrt{0.04}=0.2\)

Ответ: 0.2

Решение №16761: \(\frac{\left ( 2p-q \right )^{2}+2q^{2}-3pq}{2p^{-1}+q^{2}}:\frac{4p^{2}-3pq}{2+pq^{2}}=\frac{4p^{2}-4pq+q^{2}+2q^{2}-3pq}{\frac{2}{p}+q^{2}}\cdot \frac{2+pq^{2}}{p\left ( 4p-3q \right )}=\frac{\left ( 4p^{2}-7pq+3q^{2} \right )p}{2+pq^{2}}\cdot \frac{2+pq^{2}}{p\left ( 4p-3q \right )}=\frac{\left ( p-q \right )\left ( 4p-3q \right )}{4p-3q}=p-q=0.78-\frac{7}{25}=0.78-0.28=0.5\)

Ответ: 0.5

Решение №16762: \(\frac{a^{3}-a-2b-\frac{b^{2}}{a}}{\left ( 1-\sqrt{\frac{1}{a}+\frac{b}{a^{2}}} \right )\left ( a+\sqrt{a+b} \right )}:\left ( \frac{a^{3}+a^{2}+ab+a^{2}b}{a^{2}-b^{2}}+\frac{b}{a-b} \right )=\frac{a^{4}-\left ( a^{2}+2ab+b^{2} \right )}{\left ( a-\sqrt{a+b} \right )\left ( a+\sqrt{a+b} \right )}:\left ( \frac{a\left ( a+1 \right )\left ( a+b \right )}{\left ( a-1 \right )\left ( a+b \right )}+\frac{b}{a-b} \right )=\frac{a^{4}-\left ( a+b \right )^{2}}{a^{2}-a-b}:\left ( \frac{a\left ( a+1 \right )}{a-b}+\frac{b}{a-b} \right )=\frac{\left ( a^{2}+a+b \right )\left ( a-b \right )}{a^{2}+a+b}=a-b=23-22=1\)

Ответ: 1

Решение №16763: \(\frac{\sqrt{2}\left ( x-a \right )}{2x-a}-\left ( \left ( \frac{\sqrt{x}}{\sqrt{2x}+\sqrt{a}} \right )^{2}+\left ( \frac{\sqrt{2x}+\sqrt{a}}{2\sqrt{a}} \right )^{-1} \right )^{\frac{1}{2}}=\frac{\sqrt{2}\left ( x-a \right )}{2x-a}-\left ( \frac{x}{\left ( \sqrt{2x}+\sqrt{a} \right )}+\frac{2\sqrt{a}}{\sqrt{2x}+\sqrt{a}} \right )^{\frac{1}{2}}=\frac{\sqrt{2}\left ( x-a \right )}{\left ( \sqrt{2x}-\sqrt{a} \right )\left ( \sqrt{2x}+\sqrt{a} \right )}-\frac{\sqrt{x}+\sqrt{2a}}{\sqrt{2x}+\sqrt{a}}=\frac{x\sqrt{2}-a\sqrt{2}-x\sqrt{2}+\sqrt{ax}-2\sqrt{ax}+a\sqrt{2}}{2x-a}=\frac{-\sqrt{ax}}{2x-a}=\frac{-\sqrt{0.32\cdot 0.08}}{2\cdot 0.08-0.32}=\frac{-0.16}{-0.16}=1\)

Ответ: 1

Решение №16764: \(\left ( \frac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}+\frac{x-a}{\sqrt{x^{2}-a^{2}}-x+a} \right ):\sqrt{\frac{x^{2}}{a^{2}}-1}=\left ( \frac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}+\frac{\left ( \sqrt{x-a} \right )^{2}}{\sqrt{x-a}\left ( \sqrt{x+a}-\sqrt{x-a} \right )} \right ):\sqrt{\frac{x^{2}-a^{2}}{a^{2}}}=\left ( \frac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}+\frac{\sqrt{x-a}}{\sqrt{x+a}-\sqrt{x-a}} \right )\cdot \frac{a}{\sqrt{x^{2}-a^{2}}}=\frac{\sqrt{x^{2}-a^{2}}-x+a+\sqrt{x^{2}-a^{2}}+x-a}{x+a-x+a}\cdot \frac{a}{\sqrt{x^{2}-a^{2}}}=\frac{2a\sqrt{x^{2}-a^{2}}}{2a\sqrt{x^{2}-a^{2}}}=1\)

Ответ: 1

Решение №16765: \(\frac{3a^{2}+2ax-x^{2}}{\left ( 3x+a \right )\left ( a+x \right )}-2+10\cdot \frac{ax-3x^{2}}{a^{2}-9x^{2}}=\frac{-\left ( x+a \right )\left ( x-3a \right )}{\left ( 3x+a \right )\left ( a+x \right )}-2+10\cdot \frac{x\left ( a-3x \right )}{\left ( a-3x \right )\left ( a+3x \right )}=\frac{-x+3a}{3x+a}-2+\frac{10x}{3x+a}=\frac{-x+3a-6x-2a+10x}{3x+a}=1\)

Ответ: 1

Решение №16766: \(\frac{\sqrt{5-2\sqrt{6}}}{\left ( \sqrt[4]{3}+\sqrt[4]{2} \right )\left ( \sqrt[4]{3}-\sqrt[4]{2} \right )}=\frac{\sqrt{3-2\sqrt{3\cdot 2}+2}}{\left ( \sqrt[4]{3} \right )^{2}-\left ( \sqrt[4]{2} \right )^{2}}=\frac{\sqrt{\left ( \sqrt{3}-\sqrt{2} \right )^{2}}}{\sqrt{3}-\sqrt{2}}=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}=1\)

Ответ: 1

Решение №16879: \(\left ( \frac{a\sqrt{a}+b\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\sqrt{ab} \right )\cdot \left ( \frac{\sqrt{a}+\sqrt{b}}{a-b} \right )^{2}=\left ( \frac{\sqrt{a^{3}}+\sqrt{b^{3}}}{\sqrt{a}+\sqrt{b}}-\sqrt{ab} \right )\cdot \left ( \frac{\sqrt{a}+\sqrt{b}}{\left ( \sqrt{a} \right )^{2}-\left ( \sqrt{b} \right )^{2}} \right )^{2}=\left ( \sqrt{a^{2}}-2\sqrt{ab}+\sqrt{b^{2}} \right )\frac{1}{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}=\frac{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}{\left ( \sqrt{a}-\sqrt{b} \right )^{2}}=1\)

Ответ: 1

Решение №16880: \(\frac{1-\sqrt{2t}}{\frac{1-\sqrt[4]{8t^{3}}}{1-\sqrt[4]{2t}}-\sqrt{2t}}\cdot \left ( \frac{\sqrt[4]{\frac{1}{2t}}+\sqrt[4]{4t^{2}}}{1+\sqrt[4]{\frac{1}{2t}}}-\sqrt{2t} \right )^{-1}=\frac{\left ( 1-\sqrt[4]{4t^{2}} \right )\left ( 1-\sqrt[4]{2t} \right )}{1-\sqrt[4]{4t^{2}}}\cdot \left ( \frac{1+\sqrt[4]{8t^{3}}}{1+\sqrt[4]{2t}}-\sqrt[4]{4t^{2}} \right )^{-1}=\left ( 1-\sqrt[4]{2t} \right )\left ( 1-\sqrt[4]{2t}+\sqrt[4]{4t^{2}}-\sqrt[4]{4t^{2}} \right )^{-1}=\left ( 1-\sqrt[4]{2t} \right )\left ( 1-\sqrt[4]{2t} \right )^{-1}=\frac{\left ( 1-\sqrt[4]{2t} \right )}{\left ( 1-\sqrt[4]{2t} \right )}=1\)

Ответ: 1

Решение №16881: \(\frac{1-ax}{1+ax}\sqrt{\frac{1+bx}{1-bx}}; x=\frac{1}{a}\sqrt{\frac{2a-b}{b}}; = \frac{1-a\frac{1}{a}\sqrt{\frac{2a-b}{b}}}{1+a\frac{1}{a}\sqrt{\frac{2a-b}{b}}}\cdot \sqrt{\frac{1+b\frac{1}{a}\sqrt{\frac{2a-b}{2}}}{1-b\frac{1}{a}\sqrt{\frac{2a-b}{b}}}}=\frac{1-\sqrt{\frac{2a-b}{}}}{1+\sqrt{\frac{2a-b}{b}}}\cdot \sqrt{\frac{1+\frac{1}{a}\sqrt{\frac{b^{2}\left ( 2a-b \right )}{b}}}{1-\frac{1}{a}\sqrt{\frac{b^{2}\left ( 2a-b \right )}{b}}}}=\frac{a-\sqrt{b\left ( 2a-b \right )}}{b-a}\sqrt{\left ( \frac{a+\sqrt{b}\left ( 2a-b \right )}{a-b} \right )^{2}}=\frac{a^{2}-b\left ( 2a-b \right )}{\left ( b-a \right )^{2}}=\frac{\left ( b-a \right )^{2}}{\left ( b-a \right )^{2}}=1\)

Ответ: 1

Решение №16882: \(\frac{\left ( a^{2}b\sqrt{b}-6a^{\frac{5}{3}}b^{\frac{5}{4}}+12ab\sqrt[3]{a}-8ab^{\frac{3}{4}} \right )^{\frac{2}{3}}}{ab\sqrt[3]{a}-4ab^{\frac{3}{4}}+4a^{\frac{2}{3}}\sqrt{b}}=\frac{\left ( a^{2}b^{\frac{3}{2}}-6a^{\frac{5}{3}}b^{\frac{5}{4}}+12a^{\frac{4}{3}}b-8ab^{\frac{3}{4}} \right )^{\frac{2}{3}}}{a^{\frac{4}{3}}b-4ab^{\frac{3}{4}}+4a^{\frac{2}{3}}b^{\frac{1}{2}}}=\frac{a^{\frac{2}{3}}b^{\frac{1}{2}}\left ( \left ( ab^{\frac{3}{4}}-8 \right )-6a^{\frac{1}{3}}b^{\frac{1}{4}}\left ( ^{\frac{1}{3}}b^{\frac{1}{4}}-2 \right ) \right )^{\frac{2}{3}}}{a^{\frac{2}{3}}b^{\frac{1}{2}}\left ( a^{\frac{1}{3}}b^{\frac{1}{4}}-2 \right )^{2}}=\frac{\left ( \left ( a^{\frac{1}{3}}b^{\frac{1}{4}}-2 \right )\left ( a^{\frac{1}{3}}b^{\frac{1}{4}}-2 \right )^{2} \right )^{\frac{2}{3}}}{\left (a^{\frac{1}{3}}b^{\frac{1}{4}}-2 \right )^{2}}=\frac{\left ( \left ( a^{\frac{1}{3}}b^{\frac{1}{4}}-2 \right )^{3} \right )^{\frac{2}{3}}}{\left ( a^{\frac{1}{3}}b^{\frac{1}{4}}-2 \right )^{2}}=\frac{\left ( a^{\frac{1}{3}}b^{\frac{1}{4}}-2 \right )^{2}}{\left ( a^{\frac{1}{3}}b^{\frac{1}{4}}-2 \right )^{2}}=1\)

Ответ: 1

Решение №16883: \(\sqrt{2+\sqrt{3}}\sqrt{2+\sqrt{2+\sqrt{3}}}\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}=\sqrt{2+\sqrt{3}}\sqrt{2+\sqrt{2+\sqrt{3}}}\sqrt{4-2-\sqrt{2+\sqrt{3}}}=\sqrt{2+\sqrt{3}}\sqrt{2+\sqrt{2+\sqrt{3}}}\sqrt{2-\sqrt{2+\sqrt{3}}}=\sqrt{2+\sqrt{3}}\sqrt{4-2-\sqrt{3}}=\sqrt{2+\sqrt{3}}\sqrt{2-\sqrt{3}}=\sqrt{\left ( 2+\sqrt{3} \right )\left ( 2-\sqrt{3} \right )}=\sqrt{4-3}=1\)

Ответ: 1