Solve {l}{x^6-y^6}{x^12-y^6}

$$\left. \begin{array} { l } { x ^ { 6 } - y ^ { 6 } } \\ { x ^ { 12 } - y ^ { 6 } } \end{array} \right.$$

Least Common Multiple

$$\left(x^{2}-y^{2}\right)\left(x^{4}-y^{2}\right)\left(-\left(xy\right)^{2}+\left(x^{2}+y^{2}\right)^{2}\right)\left(\left(x^{4}+y^{2}\right)^{2}-\left(yx^{2}\right)^{2}\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$x^{6}-y^{6}=\left(x+y\right)\left(x-y\right)\left(x^{2}-xy+y^{2}\right)\left(x^{2}+xy+y^{2}\right) x^{12}-y^{6}=\left(x^{2}+y\right)\left(x^{2}-y\right)\left(x^{4}+y^{2}-yx^{2}\right)\left(x^{4}+y^{2}+yx^{2}\right)$$

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

$$\left(x-y\right)\left(x+y\right)\left(x^{2}-y\right)\left(x^{2}+y\right)\left(x^{2}-xy+y^{2}\right)\left(x^{2}+xy+y^{2}\right)\left(x^{4}+y^{2}-yx^{2}\right)\left(x^{4}+y^{2}+yx^{2}\right)$$

Expand the expression.

$$x^{18}-x^{6}y^{6}+y^{12}-y^{6}x^{12}$$

Evaluate

$$\left(x^{2}-y^{2}\right)\left(-\left(xy\right)^{2}+\left(x^{2}+y^{2}\right)^{2}\right),\ \left(x^{4}-y^{2}\right)\left(\left(x^{4}+y^{2}\right)^{2}-\left(yx^{2}\right)^{2}\right)$$