Solve {l}{4x^{6}y^{3}+y^{15}}{x^{2}(a+b)^2-2xy(a^2-b^2)+v^2}

$$\left. \begin{array} { l } { 4 x ^ { 6 } y ^ { 3 } + y ^ { 15 } } \\ { x ^ { 2 } ( a + b ) ^ { 2 } - 2 x y ( a ^ { 2 } - b ^ { 2 } ) + v ^ { 2 } } \end{array} \right.$$

Least Common Multiple

$$lcm(y^{3}\left(4x^{6}+y^{12}\right),\left(x\left(a+b\right)\right)^{2}-2xy\left(a^{2}-b^{2}\right)+v^{2})$$

Solution Steps

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

$$y^{3}\left(a^{2}x^{2}+b^{2}x^{2}+2abx^{2}+2xyb^{2}-2xya^{2}+v^{2}\right)\left(4x^{6}+y^{12}\right)$$

Expand the expression.

$$a^{2}x^{2}y^{15}+b^{2}x^{2}y^{15}+2abx^{2}y^{15}+2xb^{2}y^{16}-2xa^{2}y^{16}+v^{2}y^{15}+8b^{2}y^{4}x^{7}-8a^{2}y^{4}x^{7}+8aby^{3}x^{8}+4a^{2}y^{3}x^{8}+4b^{2}y^{3}x^{8}+4v^{2}y^{3}x^{6}$$

Evaluate

$$y^{3}\left(4x^{6}+y^{12}\right),\ \left(x\left(a+b\right)\right)^{2}-2xy\left(a^{2}-b^{2}\right)+v^{2}$$