# Perform the indicated operation and express all answers with a rational denominator 2a^{2}bsqrt[3]{4a^{3}b} cdot -6ab^{2}sqrt[3]{18ab^{4}}

Question
Perform the indicated operation and express all answers with a rational denominator $$2a^{2}b\sqrt[3]{4a^{3}b}\ \cdot\ -6ab^{2}\sqrt[3]{18ab^{4}}$$

2021-02-22
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: $$-12a^{3}b^{5}\sqrt[3]{72a^{4}b^{5}}$$

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