Заметим формулы сокращенного умножения:
\(\displaystyle\overset{\color{red}{\text квадрат\ суммы}}{\color{blue}{(n^3+m^3)^2}}-\overset{\color{red}{\text куб\ суммы}}{\color{green}{(n^2+m^2)^3}}+3n^2m^2\cdot\overset{\color{red}{\text квадрат\ суммы}}{\color{magenta}{(n+m)^2}}-8n^3m^3{\small.}\)
Применим формулу квадрата суммы:\(\displaystyle (n^3+m^3)^2=n^6+2n^3m^3+m^6{\small.}\)
ПравилоКвадрат суммы
Для любых чисел \(\displaystyle a\) и \(\displaystyle b\) верно
\(\displaystyle (a+b)^{\,2}=a\,^2+2ab+b\,^2{\small.}\)
В нашем случае \(\displaystyle a=\color{blue}{n^3}{\small,}\) \(\displaystyle b=\color{brown}{m^3}{\small.}\) Получаем:
\(\displaystyle (\color{blue}{n^3}+\color{brown}{m^3})^2=(\color{blue}{n^3})^2+2\cdot\color{blue}{n^3}\cdot\color{brown}{m^3}+(\color{brown}{m^3})^2=n^6+2n^3m^3+m^6{\small.}\)
Применим формулу куба суммы: \(\displaystyle (n^2+m^2)^3=n^6+3n^4m^2+3n^2m^4+m^6{\small.}\)
ПравилоКуб суммы
Для любых чисел \(\displaystyle a\) и \(\displaystyle b\) верно
\(\displaystyle (a+b\,)^3=a^{\,3}+3a^{\,2}b+3ab^{\,2}+b^{\,3}{\small.}\)
В нашем случае \(\displaystyle a=\color{magenta}{n^2} {\small,}\) \(\displaystyle b=\color{orange}{m^2}{\small.}\) Получаем:
\(\displaystyle \begin{aligned}(\color{magenta}{n^2}+\color{orange}{m^2})^3&=(\color{magenta}{n^2})^3+3\cdot (\color{magenta}{n^2})^2 \cdot \color{orange}{m^2}+3\cdot \color{magenta}{n^2}\cdot (\color{orange}{m^2})^2+(\color{orange}{m^2})^3=\\&=n^6+3n^4m^2+3n^2m^4+m^6{\small.}\end{aligned}\)
Применим формулу квадрата суммы:\(\displaystyle (n+m)^2=n^2+2nm+m^2{\small.}\)
ПравилоКвадрат суммы
Для любых чисел \(\displaystyle a\) и \(\displaystyle b\) верно
\(\displaystyle (a+b)^{\,2}=a\,^2+2ab+b\,^2{\small.}\)
В нашем случае \(\displaystyle a=\color{blue}{n}{\small,}\) \(\displaystyle b=\color{brown}m{\small.}\) Получаем:
\(\displaystyle (\color{blue}{n}+\color{brown}m)^2=\color{blue}{n}^2+2\cdot\color{blue}{n}\cdot\color{brown}m+\color{brown}m^2=n^2+2nm+m^2{\small.}\)
Получаем:
\(\displaystyle \begin{aligned}\color{blue}{(n^3+m^3)^2}-\color{green}{(n^2+m^2)^3}&+3n^2m^2\cdot\color{magenta}{(n+m)^2}-8n^3m^3=\\=\color{blue}{(n^6+2n^3m^3+m^6)}&- \color{green}{(n^6+3n^4m^2+3n^2m^4+m^6)}+\\&+3n^2m^2 \cdot \color{magenta}{(n^2+2nm+m^2)}-8n^3m^3{\small.}\end{aligned}\)
Раскроем скобки и приведём подобные слагаемые:
\(\displaystyle \begin{aligned}(n^6+2n^3m^3+m^6)&- (n^6+3n^4m^2+3n^2m^4+m^6)+\\&+3n^2m^2 \cdot (n^2+2nm+m^2)-8n^3m^3=\\=n^6+2n^3m^3+m^6&- n^6-3n^4m^2-3n^2m^4-m^6+\\&+3n^4m^2+6n^3m^3+3n^2m^4-8n^3m^3=\\=\color{magenta}{\cancel{n^6}}+\color{orange}{\cancel{2n^3m^3}}+\color{brown}{\cancel{m^6}}&- \color{magenta}{\cancel{n^6}}-\color{RoyalBlue}{\cancel{3n^4m^2}}-\color{purple}{\cancel{3n^2m^4}}-\color{brown}{\cancel{m^6}}+\\&+\color{RoyalBlue}{\cancel{3n^4m^2}}+\color{orange}{\cancel{6n^3m^3}}+\color{purple}{\cancel{3n^2m^4}}-\color{orange}{\cancel{8n^3m^3}}=0\, \, {\small.}\end{aligned}\)
Ответ: \(\displaystyle 0{\small.}\)