Question

Simplify $a^6-b^6.$

Answer 1

Given expression: $a^6-b^6.$

$\begin{array}{l}\begin{array}{c}Thegivenexpressionis{\mathbf{a}}^6-{\mathbf{b}}^6.\\ {\mathbf{a}}^6-{\mathbf{b}}^6={\left({\mathbf{a}}^2\right)}^3-{\left({\mathbf{b}}^2\right)}^3\\ ({\mathbf{a}}^3-{\mathbf{b}}^3)=(\mathbf{a}-\mathbf{b})({\mathbf{a}}^2+\mathbf{ab}+{\mathbf{b}}^2)\\ =({\mathbf{a}}^2-{\mathbf{b}}^2)[{\left({\mathbf{a}}^2\right)}^2+{\mathbf{a}}^2{b}^2+{\left({\mathrm{b}}^2\right)}^2]\\ =(a+b)(a-b)[{\left(a^2\right)}^2+2a^2{b}^2+{\left(b^2\right)}^2-a^2{b}^2]\\ =(a+b)(a-b)[{(a^2+b^2)}^2-a^2{b}^2]\\ =(\mathbf{a}+\mathbf{b})(\mathbf{a}-\mathbf{b})\left[({\mathbf{a}}^2+{\mathbf{b}}^2+\mathbf{ab})({\mathbf{a}}^2+{\mathbf{b}}^2-\mathbf{ab})\right]\end{array}\end{array}$