Question

$x^{12}-y^{12}$ is equal to ?

Answer 1

Simplify using algebraic identities

$$\begin{gathered} x^{12}-y^{12}={\left(x^6\right)}^2-{\left(y^6\right)}^2 \\ =\left(x^6-y^6\right)\left(x^6+y^6\right)\left[\because a^2-b^2=\left(a-b\right)\left(a+b\right)\right] \\ =\left({\left(x^3\right)}^2-{\left(y^3\right)}^2\right)\left(x^6+y^6\right) \\ =\left(x^3-y^3\right)\left(x^3+y^3\right)\left(x^6+y^6\right)\left[\because a^2-b^2=\left(a-b\right)\left(a+b\right)\right] \\ =\left(x^3-y^3\right)\left(x^3+y^3\right)\left(x^6+y^6\right) \\ =\left(x-y\right)\left(x^2+y^2+xy\right)\left(x^3+y^3\right)\left(x^6+y^6\right)\left[\because a^3-b^3=\left(a-b\right)\left(a^2+b^2+ab\right)\right] \end{gathered}$$

Thus, $\left(x^{12}-y^{12}\right)=\left(x-y\right)\left(x^2+y^2+xy\right)\left(x^3+y^3\right)\left(x^6+y^6\right)$