Question - Simplification of Radical Expressions with Negative Exponent
Solution:
$$(3 \sqrt[3]{m^6n^3})(4 \sqrt[3]{m^6n^2})^{-\frac{1}{2}} = 3 \sqrt[3]{m^6n^3} \cdot (4 \sqrt[3]{m^6n^2})^{-\frac{1}{2}}$$$$= 3 \sqrt[3]{m^6n^3} \cdot \left( \frac{1}{4 \sqrt[3]{m^6n^2}} \right)^{\frac{1}{2}}$$$$= 3 \sqrt[3]{m^6n^3} \cdot \left( \frac{1}{2 \sqrt[3]{m^3n}} \right)$$$$= \frac{3}{2} \cdot \frac{\sqrt[3]{m^6n^3}}{\sqrt[3]{m^3n}}$$$$= \frac{3}{2} \sqrt[3]{\frac{m^6n^3}{m^3n}}$$$$= \frac{3}{2} \sqrt[3]{m^3n^2}$$$$= \frac{3}{2}mn^{\frac{2}{3}}$$
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