Калькуляторы онлайн/ Решение уравнений/ Упрощение выражений/ Общий множитель за скобки/ 12*a^2*b^2+18*a^5-16*a*b^6

Общий множитель 12*a^2*b^2+18*a^5-16*a*b^6

Учитель очень удивится увидев твоё верное решение 😼

Примеры

Решение

Комбинаторика [src]
    /     6      4        2\
2*a*\- 8*b  + 9*a  + 6*a*b /
2a(9a4+6ab2−8b6)2 a \left(9 a^{4} + 6 a b^{2} - 8 b^{6}\right)
Разложение на множители [src]
  /    //                                                                                                                    _____________________________________                                                                                                                                                     \\ /    //                                                                                                                  _____________________________________                                                                                                                                                     \\ /    //                                                                                                                   _____________________________________                                                                                                                                                    \\ /    //                                                                                                                  _____________________________________                                                                                                                                                     \\
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  |    ||                                                                                                                  /  3 ____ 3 /   4        2/3 3 ___  2                                                                                                                                                       || |    ||                                                                                                                /  3 ____ 3 /   4        2/3 3 ___  2                                                                                                                                                       || |    ||                                                                                                                 /  3 ____ 3 /   4        2/3 3 ___  2                                                                                                                                                      || |    ||                                                                                                                /  3 ____ 3 /   4        2/3 3 ___  2                                                                                                                                                       ||
  |    ||                                                                                                                 /   \/ 12 *\/  -b      2*2   *\/ 3 *b                                                                                                                                                        || |    ||                                                                                                               /   \/ 12 *\/  -b      2*2   *\/ 3 *b                                                                                                                                                        || |    ||                                                                                                                /   \/ 12 *\/  -b      2*2   *\/ 3 *b                                                                                                                                                       || |    ||                                                                                                               /   \/ 12 *\/  -b      2*2   *\/ 3 *b                                                                                                                                                        ||
  |    ||                                                                                                                /    --------------- + -----------------                                                                                                                                                      || |    ||                                                                                                              /    --------------- + -----------------                                                                                                                                                      || |    ||                                                                                                               /    --------------- - -----------------                                                                                                                                                     || |    ||                                                                                                              /    --------------- - -----------------                                                                                                                                                      ||
  |    ||                                                                                                               /            3                ___________                   ___________                                                                                                                        || |    ||                                                                                                             /            3                ___________                   ___________                                                                                                                        || |    ||                                                                                                              /            3                ___________                   ___________                                                                                                                       || |    ||                                                                                                             /            3                ___________                   ___________                                                                                                                        ||
  |    ||                                                                                                              /                             /     _____                   /     _____                                                                                                                         || |    ||                                                                                                            /                             /     _____                   /     _____                                                                                                                         || |    ||                                                                                                             /                             /     _____                   /     _____                                                                                                                        || |    ||                                                                                                            /                             /     _____                   /     _____                                                                                                                         ||
  |    ||                                                                                                             /                             /   3 /   4      3 ___  2/3   /   3 /   4                                                                                                                     6    || |    ||                                                                                                           /                             /   3 /   4      3 ___  2/3   /   3 /   4                                                                                                                     6    || |    ||                                                                                                            /                             /   3 /   4      3 ___  2/3   /   3 /   4                                                                                                                    6    || |    ||                                                                                                           /                             /   3 /   4      3 ___  2/3   /   3 /   4                                                                                                                     6    ||
  |    ||                                                                                                           \/                          3*\/   -\/  -b       \/ 2 *3   *\/   -\/  -b                                                                                                                   8*b     || |    ||                                                                                                         \/                          3*\/   -\/  -b       \/ 2 *3   *\/   -\/  -b                                                                                                                   8*b     || |    ||                                                                                                          \/                          3*\/   -\/  -b       \/ 2 *3   *\/   -\/  -b                                                                                                                  8*b     || |    ||                                                                                                         \/                          3*\/   -\/  -b       \/ 2 *3   *\/   -\/  -b                                                                                                                   8*b     ||
  |    ||                                                                                                         - ---------------------------------------------- - --------------------------                                                                                                            for ---- = 0|| |    ||                                                                                                         ---------------------------------------------- - --------------------------                                                                                                            for ---- = 0|| |    ||                                                                                                        - ---------------------------------------------- + --------------------------                                                                                                           for ---- = 0|| |    ||                                                                                                         ---------------------------------------------- + --------------------------                                                                                                            for ---- = 0||
  |    ||                                                                                                                                 2                                      6                                                                                                                              9      || |    ||                                                                                                                               2                                      6                                                                                                                              9      || |    ||                                                                                                                                2                                      6                                                                                                                             9      || |    ||                                                                                                                               2                                      6                                                                                                                              9      ||
  |    ||                                                                                                                                                                                                                                                                                                              || |    ||                                                                                                                                                                                                                                                                                                            || |    ||                                                                                                                                                                                                                                                                                                            || |    ||                                                                                                                                                                                                                                                                                                            ||
  |    ||                                                                                                                 _______________________________________________________________________________________________________________________________________________________________________________              || |    ||                  _______________________________________________________________________________________________________________________________________________________________________________                                                                                                           || |    ||                                                                                                               _______________________________________________________________________________________________________________________________________________________________________________              || |    ||                                                                                                               _______________________________________________________________________________________________________________________________________________________________________________              ||
  |    ||                                                                                                                /           ____________________________                                                                                                                                                      || |    ||                 /           ____________________________                                                                                                                                                                                                                                                   || |    ||                                                                                                              /           ____________________________                                                                                                                                                      || |    ||                                                                                                              /           ____________________________                                                                                                                                                      ||
  |    ||                                                                                                               /           /      ________________                                                                                                                                                            || |    ||                /           /      ________________                                                                                                                                                                                                                                                         || |    ||                                                                                                             /           /      ________________                                                                                                                                                            || |    ||                                                                                                             /           /      ________________                                                                                                                                                            ||
  |    ||                                                                                                              /           /      /   8         18     4                                                   2                                                                   6                               || |    ||               /           /      /   8         18     4                                                   2                                                                   6                                                                                                                            || |    ||                                                                                                            /           /      /   8         18     4                                                   2                                                                   6                               || |    ||                                                                                                            /           /      /   8         18     4                                                   2                                                                   6                               ||
a*|a - |<                                                                                                             /           /      /   b     512*b      b                                                 4*b                                                                16*b                                ||*|a - |<              /           /      /   b     512*b      b                                                 4*b                                                                16*b                                                                                                                             ||*|a - |<                                                                                                           /           /      /   b     512*b      b                                                 4*b                                                                16*b                                ||*|a - |<                                                                                                           /           /      /   b     512*b      b                                                 4*b                                                                16*b                                ||
  |    ||                                                                                                            /     - 2*3 /      /   ---- + -------  + --  + -------------------------------------------------------------------------------------------- + -------------------------------------               || |    ||             /     - 2*3 /      /   ---- + -------  + --  + -------------------------------------------------------------------------------------------- + -------------------------------------                                                                                                            || |    ||                                                                                                          /     - 2*3 /      /   ---- + -------  + --  - -------------------------------------------------------------------------------------------- + -------------------------------------               || |    ||                                                                                                          /     - 2*3 /      /   ---- + -------  + --  - -------------------------------------------------------------------------------------------- + -------------------------------------               ||
  |    ||              ______________________________________________________________________________               /          \/     \/    1296    19683     36                  ______________________________________________________________________________            ____________________________               || |    ||            /          \/     \/    1296    19683     36                  ______________________________________________________________________________            ____________________________                ______________________________________________________________________________              || |    ||            ______________________________________________________________________________               /          \/     \/    1296    19683     36                  ______________________________________________________________________________            ____________________________               || |    ||            ______________________________________________________________________________               /          \/     \/    1296    19683     36                  ______________________________________________________________________________            ____________________________               ||
  |    ||             /         ____________________________                                                       /                                                             /         ____________________________                                                    /      ________________                     || |    ||           /                                                             /         ____________________________                                                    /      ________________                     /         ____________________________                                                       || |    ||           /         ____________________________                                                       /                                                             /         ____________________________                                                    /      ________________                     || |    ||           /         ____________________________                                                       /                                                             /         ____________________________                                                    /      ________________                     ||
  |    ||            /         /      ________________                                                            /                                                             /         /      ________________                                                         /      /   8         18     4                || |    ||          /                                                             /         /      ________________                                                         /      /   8         18     4               /         /      ________________                                                             || |    ||          /         /      ________________                                                            /                                                             /         /      ________________                                                         /      /   8         18     4                || |    ||          /         /      ________________                                                            /                                                             /         /      ________________                                                         /      /   8         18     4                ||
  |    ||           /         /      /   8         18     4                        6                             /                                                             /         /      /   8         18     4                        6                          /      /   b     512*b      b                 || |    ||         /                                                             /         /      /   8         18     4                        6                          /      /   b     512*b      b               /         /      /   8         18     4                        6                               || |    ||         /         /      /   8         18     4                        6                             /                                                             /         /      /   8         18     4                        6                          /      /   b     512*b      b                 || |    ||         /         /      /   8         18     4                        6                             /                                                             /         /      /   8         18     4                        6                          /      /   b     512*b      b                 ||
  |    ||          /         /      /   b     512*b      b                     16*b                             /                                                             /         /      /   b     512*b      b                     16*b                     27*3 /      /   ---- + -------  + --                || |    ||        /                                                             /         /      /   b     512*b      b                     16*b                     27*3 /      /   ---- + -------  + --             /         /      /   b     512*b      b                     16*b                                || |    ||        /         /      /   b     512*b      b                     16*b                             /                                                             /         /      /   b     512*b      b                     16*b                     27*3 /      /   ---- + -------  + --                || |    ||        /         /      /   b     512*b      b                     16*b                             /                                                             /         /      /   b     512*b      b                     16*b                     27*3 /      /   ---- + -------  + --                ||
  |    ||         /     2*3 /      /   ---- + -------  + --  - -------------------------------------           /                                                    3*       /     2*3 /      /   ---- + -------  + --  - -------------------------------------       \/     \/    1296    19683     36                || |    ||       /                                                    3*       /     2*3 /      /   ---- + -------  + --  - -------------------------------------       \/     \/    1296    19683     36            /     2*3 /      /   ---- + -------  + --  - -------------------------------------               || |    ||       /     2*3 /      /   ---- + -------  + --  - -------------------------------------           /                                                    3*       /     2*3 /      /   ---- + -------  + --  - -------------------------------------       \/     \/    1296    19683     36                || |    ||       /     2*3 /      /   ---- + -------  + --  - -------------------------------------           /                                                    3*       /     2*3 /      /   ---- + -------  + --  - -------------------------------------       \/     \/    1296    19683     36                ||
  |    ||        /        \/     \/    1296    19683     36             ____________________________          /                                                             /        \/     \/    1296    19683     36             ____________________________                                                        || |    ||      /                                                             /        \/     \/    1296    19683     36             ____________________________                                                   /        \/     \/    1296    19683     36             ____________________________               || |    ||      /        \/     \/    1296    19683     36             ____________________________          /                                                             /        \/     \/    1296    19683     36             ____________________________                                                        || |    ||      /        \/     \/    1296    19683     36             ____________________________          /                                                             /        \/     \/    1296    19683     36             ____________________________                                                        ||
  |    ||       /                                                      /      ________________               /                                                             /                                                      /      ________________                                                              || |    ||     /                                                             /                                                      /      ________________                                                        /                                                      /      ________________                     || |    ||     /                                                      /      ________________               /                                                             /                                                      /      ________________                                                              || |    ||     /                                                      /      ________________               /                                                             /                                                      /      ________________                                                              ||
  |    ||      /                                                      /      /   8         18     4         /                                                             /                                                      /      /   8         18     4                                                         || |    ||    /                                                             /                                                      /      /   8         18     4                                                  /                                                      /      /   8         18     4                || |    ||    /                                                      /      /   8         18     4         /                                                             /                                                      /      /   8         18     4                                                         || |    ||    /                                                      /      /   8         18     4         /                                                             /                                                      /      /   8         18     4                                                         ||
  |    ||     /                                                      /      /   b     512*b      b         /                                                             /                                                      /      /   b     512*b      b                                                          || |    ||   /                                                             /                                                      /      /   b     512*b      b                                                  /                                                      /      /   b     512*b      b                 || |    ||   /                                                      /      /   b     512*b      b         /                                                             /                                                      /      /   b     512*b      b                                                          || |    ||   /                                                      /      /   b     512*b      b         /                                                             /                                                      /      /   b     512*b      b                                                          ||
  |    ||    /                                                 27*3 /      /   ---- + -------  + --       /                                                             /                                                 27*3 /      /   ---- + -------  + --                                                         || |    ||  /                                                             /                                                 27*3 /      /   ---- + -------  + --                                                /                                                 27*3 /      /   ---- + -------  + --                || |    ||  /                                                 27*3 /      /   ---- + -------  + --       /                                                             /                                                 27*3 /      /   ---- + -------  + --                                                         || |    ||  /                                                 27*3 /      /   ---- + -------  + --       /                                                             /                                                 27*3 /      /   ---- + -------  + --                                                         ||
  |    ||  \/                                                     \/     \/    1296    19683     36     \/                                                            \/                                                     \/     \/    1296    19683     36                                                         || |    ||\/                                                            \/                                                     \/     \/    1296    19683     36                                              \/                                                     \/     \/    1296    19683     36                || |    ||\/                                                     \/     \/    1296    19683     36     \/                                                            \/                                                     \/     \/    1296    19683     36                                                         || |    ||\/                                                     \/     \/    1296    19683     36     \/                                                            \/                                                     \/     \/    1296    19683     36                                                         ||
  |    ||- ------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------   otherwise  || |    ||------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------   otherwise  || |    ||------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------   otherwise  || |    ||------------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------   otherwise  ||
  |    ||                                              2                                                                                                                                                2                                                                                                              || |    ||                                                                                                2                                                                                                                                               2                                                           || |    ||                                            2                                                                                                                                                2                                                                                                              || |    ||                                            2                                                                                                                                                2                                                                                                              ||
  \    \\                                                                                                                                                                                                                                                                                                              // \    \\                                                                                                                                                                                                                                                                                                            // \    \\                                                                                                                                                                                                                                                                                                            // \    \\                                                                                                                                                                                                                                                                                                            //
a(a−{−23⋅323−−b436−2⋅22333b23−−b43+123−b4332for 8b69=0−−16b627b436+512b1819683+b812963+2b436+512b1819683+b8129632−16b627b436+512b1819683+b812963+4b23−16b627b436+512b1819683+b812963+2b436+512b1819683+b812963−2b436+512b1819683+b8129632otherwise)(a−{−23⋅323−−b436+2⋅22333b23−−b43+123−b4332for 8b69=0−−16b627b436+512b1819683+b812963+2b436+512b1819683+b8129632+16b627b436+512b1819683+b812963+4b23−16b627b436+512b1819683+b812963+2b436+512b1819683+b812963−2b436+512b1819683+b8129632otherwise)(a−{23⋅323−−b436−−2⋅22333b23−−b43+123−b4332for 8b69=0−16b627b436+512b1819683+b812963+2b436+512b1819683+b8129632−16b627b436+512b1819683+b812963−4b23−16b627b436+512b1819683+b812963+2b436+512b1819683+b812963−2b436+512b1819683+b8129632otherwise)(a−{23⋅323−−b436+−2⋅22333b23−−b43+123−b4332for 8b69=0−16b627b436+512b1819683+b812963+2b436+512b1819683+b8129632+16b627b436+512b1819683+b812963−4b23−16b627b436+512b1819683+b812963+2b436+512b1819683+b812963−2b436+512b1819683+b8129632otherwise)a \left(a - \begin{cases} - \frac{\sqrt[3]{2} \cdot 3^{\frac{2}{3}} \sqrt{- \sqrt[3]{- b^{4}}}}{6} - \frac{\sqrt{\frac{2 \cdot 2^{\frac{2}{3}} \sqrt[3]{3} b^{2}}{3 \sqrt{- \sqrt[3]{- b^{4}}}} + \frac{\sqrt[3]{12} \sqrt[3]{- b^{4}}}{3}}}{2} & \text{for}\: \frac{8 b^{6}}{9} = 0 \\- \frac{\sqrt{- \frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}}{2} - \frac{\sqrt{\frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + \frac{4 b^{2}}{3 \sqrt{- \frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}} - 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}}{2} & \text{otherwise} \end{cases}\right) \left(a - \begin{cases} - \frac{\sqrt[3]{2} \cdot 3^{\frac{2}{3}} \sqrt{- \sqrt[3]{- b^{4}}}}{6} + \frac{\sqrt{\frac{2 \cdot 2^{\frac{2}{3}} \sqrt[3]{3} b^{2}}{3 \sqrt{- \sqrt[3]{- b^{4}}}} + \frac{\sqrt[3]{12} \sqrt[3]{- b^{4}}}{3}}}{2} & \text{for}\: \frac{8 b^{6}}{9} = 0 \\- \frac{\sqrt{- \frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}}{2} + \frac{\sqrt{\frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + \frac{4 b^{2}}{3 \sqrt{- \frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}} - 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}}{2} & \text{otherwise} \end{cases}\right) \left(a - \begin{cases} \frac{\sqrt[3]{2} \cdot 3^{\frac{2}{3}} \sqrt{- \sqrt[3]{- b^{4}}}}{6} - \frac{\sqrt{- \frac{2 \cdot 2^{\frac{2}{3}} \sqrt[3]{3} b^{2}}{3 \sqrt{- \sqrt[3]{- b^{4}}}} + \frac{\sqrt[3]{12} \sqrt[3]{- b^{4}}}{3}}}{2} & \text{for}\: \frac{8 b^{6}}{9} = 0 \\\frac{\sqrt{- \frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}}{2} - \frac{\sqrt{\frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} - \frac{4 b^{2}}{3 \sqrt{- \frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}} - 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}}{2} & \text{otherwise} \end{cases}\right) \left(a - \begin{cases} \frac{\sqrt[3]{2} \cdot 3^{\frac{2}{3}} \sqrt{- \sqrt[3]{- b^{4}}}}{6} + \frac{\sqrt{- \frac{2 \cdot 2^{\frac{2}{3}} \sqrt[3]{3} b^{2}}{3 \sqrt{- \sqrt[3]{- b^{4}}}} + \frac{\sqrt[3]{12} \sqrt[3]{- b^{4}}}{3}}}{2} & \text{for}\: \frac{8 b^{6}}{9} = 0 \\\frac{\sqrt{- \frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}}{2} + \frac{\sqrt{\frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} - \frac{4 b^{2}}{3 \sqrt{- \frac{16 b^{6}}{27 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}} + 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}} - 2 \sqrt[3]{\frac{b^{4}}{36} + \sqrt{\frac{512 b^{18}}{19683} + \frac{b^{8}}{1296}}}}}{2} & \text{otherwise} \end{cases}\right)
Объединение рациональных выражений [src]
    /     6       /   2      3\\
2*a*\- 8*b  + 3*a*\2*b  + 3*a //
2a(3a(3a3+2b2)−8b6)2 a \left(3 a \left(3 a^{3} + 2 b^{2}\right) - 8 b^{6}\right)
Собрать выражение [src]
    5         6       2  2
18*a  - 16*a*b  + 12*a *b
18a5+12a2b2−16ab618 a^{5} + 12 a^{2} b^{2} - 16 a b^{6}
Общее упрощение [src]
    /     6      4        2\
2*a*\- 8*b  + 9*a  + 6*a*b /
2a(9a4+6ab2−8b6)2 a \left(9 a^{4} + 6 a b^{2} - 8 b^{6}\right)