Given information:

The expression is \(2a^{2}b\sqrt[3]{4a^{3}b}\ \cdot\ (-6ab^{2}\sqrt[3]{18ab^{4}}).\)

Calculation:

Multiply the expression with out radicals with each other.

Use the law of exponents \(a^{b}\ \cdot\ a^{c} = a^{b\ +\ c}\)

\(2a^{2}b^{3}\ \cdot\ (-6ab^{2}\sqrt[3]{18ab^{4}})\sqrt[3]{4a^{3}b}=\ -12a^{3}b^{5}(\sqrt[3]{18ab^{4}})\sqrt[3]{4a^{3}b}\)

Multiply the expression with radicals with each other and use the law of exponents \(a^{b}\ \cdot\ a^{c} = a^{b\ +\ c}\)

\(-12a^{3}b^{5}(\sqrt[3]{18ab^{4}})\sqrt[3]{4a^{3}b}=\ -12a^{3}b^{5}(\sqrt[3]{72a^{4}b^{5}})\)

Finally answer: \(-12a^{3}b^{5}\sqrt[3]{72a^{4}b^{5}}\)

The expression is \(2a^{2}b\sqrt[3]{4a^{3}b}\ \cdot\ (-6ab^{2}\sqrt[3]{18ab^{4}}).\)

Calculation:

Multiply the expression with out radicals with each other.

Use the law of exponents \(a^{b}\ \cdot\ a^{c} = a^{b\ +\ c}\)

\(2a^{2}b^{3}\ \cdot\ (-6ab^{2}\sqrt[3]{18ab^{4}})\sqrt[3]{4a^{3}b}=\ -12a^{3}b^{5}(\sqrt[3]{18ab^{4}})\sqrt[3]{4a^{3}b}\)

Multiply the expression with radicals with each other and use the law of exponents \(a^{b}\ \cdot\ a^{c} = a^{b\ +\ c}\)

\(-12a^{3}b^{5}(\sqrt[3]{18ab^{4}})\sqrt[3]{4a^{3}b}=\ -12a^{3}b^{5}(\sqrt[3]{72a^{4}b^{5}})\)

Finally answer: \(-12a^{3}b^{5}\sqrt[3]{72a^{4}b^{5}}\)