Solve {l}{[(-1/2a^{3}b)^{3}:(1/4a^{6}b^{2})+(-2ab)^{3}:(-2b^{2})][(-3ab)(-2b^{3})-(-2ab^{2})(-2b^{2})]}{(-2a)^{2}-3a(-2a)+4b(-b)^2+(-2b)^3+(-24b^3):(-2b^2)}

$$\left. \begin{array} { l } { [ ( - \frac { 1 } { 2 } a ^ { 3 } b ) ^ { 3 } : ( \frac { 1 } { 4 } a ^ { 6 } b ^ { 2 } ) + ( - 2 a b ) ^ { 3 } : ( - 2 b ^ { 2 } ) ] [ ( - 3 a b ) ( - 2 b ^ { 3 } ) - ( - 2 a b ^ { 2 } ) ( - 2 b ^ { 2 } ) ] } \\ { ( - 2 a ) ^ { 2 } - 3 a ( - 2 a ) + 4 b ( - b ) ^ { 2 } + ( - 2 b ) ^ { 3 } + ( - 24 b ^ { 3 } ) : ( - 2 b ^ { 2 } ) } \end{array} \right.$$

Least Common Multiple

$$14\left(2b^{3}-5a^{2}-6b\right)a^{4}b^{5}$$

Solution Steps

Factor the expressions that are not already factored.

$$10a^{2}-4b^{3}+12b=2\left(5a^{2}-2b^{3}+6b\right)$$

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

$$14\left(2b^{3}-5a^{2}-6b\right)a^{4}b^{5}$$

Expand the expression.

$$28a^{4}b^{8}-84a^{4}b^{6}-70b^{5}a^{6}$$

Evaluate

$$7a^{4}b^{5},\ 10a^{2}-4b^{3}+12b$$