Solve {r}{-512+a^3b^3}{8-a^9b^3}

$$\left. \begin{array} { r } { - 512 + a ^ { 3 } b ^ { 3 } } \\ { 8 - a ^ { 9 } b ^ { 3 } } \end{array} \right.$$

Least Common Multiple

$$\left(ab-8\right)\left(ba^{3}-2\right)\left(8ab+\left(ab\right)^{2}+64\right)\left(b^{2}a^{6}+2ba^{3}+4\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$-512+a^{3}b^{3}=\left(ab-8\right)\left(a^{2}b^{2}+8ab+64\right) 8-a^{9}b^{3}=\left(ba^{3}-2\right)\left(-b^{2}a^{6}-2ba^{3}-4\right)$$

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

$$\left(ab-8\right)\left(ba^{3}-2\right)\left(a^{2}b^{2}+8ab+64\right)\left(b^{2}a^{6}+2ba^{3}+4\right)$$

Expand the expression.

$$-8a^{3}b^{3}+b^{6}a^{12}-512b^{3}a^{9}+4096$$

Evaluate

$$\left(ab\right)^{3}-512,\ 8-b^{3}a^{9}$$