Решение №16084: \(\left ( z^{2}-z+1 \right ):\left ( \left ( z^{2}+\frac{1}{z^{2}} \right )^{2}+2\left ( z+\frac{1}{z} \right )^{2}-3 \right )^{\frac{1}{2}}=\left ( z^{2}-z+1 \right ):\left ( \left ( z+\frac{1}{z} \right )^{4} -2\left ( z+\frac{1}{z} \right )^{2}+1\right )^{\frac{1}{2}}=\left ( z^{2}-z+1 \right ):\sqrt{\left ( \left ( z+\frac{1}{z} \right )^{2}-1 \right )^{2}}=\frac{z^{2}-z+1}{\frac{\left ( z^{2}-z+1 \right )\left ( z^{2}-z+1 \right )}{z^{2}}}=\frac{z^{2}}{z^{2}+z+1}\)

Ответ: \(\frac{z^{2}}{z^{2}+z+1}\)

Решение №16085: \(\frac{\sqrt{x}+1}{x\sqrt{x}+x+\sqrt{x}}:\frac{1}{x^{2}-\sqrt{x}}=\frac{\sqrt{x}+1}{\sqrt{x\left ( x+\sqrt{x}+1 \right )}}\cdot \frac{\sqrt{x\left ( x\sqrt{x}-1 \right )}}{1}=\frac{\left ( \sqrt{x}+1 \right )\left ( \sqrt{x} -1\right )}{\sqrt{x}\left (x+\sqrt{x}+1 \right )\left ( \sqrt{x}-1 \right )}\cdot \frac{\sqrt{x}\left ( x\sqrt{x}-1 \right )}{1}=\frac{x-1}{\sqrt{x}\left ( x\sqrt{x}-1 \right )}\cdot \frac{\sqrt{x}\left ( x\sqrt{x}-1 \right )}{1}=x-1\)

Ответ: \(x-1\)

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 №16105: \(3\cdot a + 4\cdot b\)

Ответ: является

Решение №16106: \(5\cdot x^{2} - 3\cdot y^{2}\)

Ответ: является

Решение №16107: \(5\cdot (5\cdot x^{2} - 12\cdot y^{2})\)

Ответ: является

Решение №16108: \((a+1)\cdot (b-2)\)

Ответ: является

Решение №16109: \(5\cdot x^{2}-6\cdot x^{2}+\frac{1}{x}\)

Ответ: не является

Решение №16110: \(\frac{3\cdot a^{2}\cdot b}{4\cdot a\cdot b^{2}}\)

Ответ: не является

Решение №16111: \(\frac{b^{2}}{4}+12\cdot z^{2}-\frac{a\cdot b}{5}\)

Ответ: является

Решение №16113: \(3\cdot x^{2}+5\cdot y+\frac{7}{c}\)

Ответ: не является

Решение №16114: \(\frac{a^{8}}{4}-\frac{b^{5}}{5}+\frac{c^{4}}{7}+\frac{d^{3}}{9}\)

Ответ: является